Show that \(S\) is

(i)     not reflexive;

(ii)     symmetric;

(iii)     transitive.

Explain why there exists an element \(a \in A\) that is not related to itself.

Hence prove that there is at least one element of \(A\) that is not related to any other element of \(A\).

Prove that \(f({e_G}) = {e_H}\).

Prove that \({\text{Ker}}(f)\) is a subgroup of \(\{ G,{\text{ }} * \} \).

Represent the following set on a Venn diagram,

\(A\Delta B\), the symmetric difference of the sets \(A\) and \(B\);

Represent the following set on a Venn diagram,

\(A \cap (B \cup C)\).

For sets \(P\), \(Q\) and \(R\), verify that \(P \cup (Q\Delta R) \ne (P \cup Q)\Delta (P \cup R)\).

In the context of the distributive law, describe what the result in part (b)(i) illustrates.

Prove that \(f \circ f\) is the identity function.

Show that \(f\) is injective.

Show that \(f\) is surjective.

(i)     Write the permutation \(\alpha = \left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 3&4&6&2&1&5 \end{array}} \right)\) as a composition of disjoint cycles.

(ii)     State the order of \(\alpha \).

(i)     Write the permutation \(\beta = \left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 6&4&3&5&1&2 \end{array}} \right)\) as a composition of disjoint cycles.

(ii)     State the order of \(\beta \).

Write the permutation \(\alpha  \circ \beta \) as a composition of disjoint cycles.

Write the permutation \(\beta  \circ \alpha \) as a composition of disjoint cycles.

State the order of \(\{ G,{\text{ }} \circ \} \).

Find the number of permutations in \(\{ G,{\text{ }} \circ \} \) which will result in \(A\), \(B\) and \(A \cap B\) remaining unchanged.

\( \odot \) is commutative;

\( * \) is associative;

\( * \) is distributive over \( \odot \) ;

\( \odot \) has an identity element.

Let \(\{ G,{\text{ }} * \} \) be a finite group that contains an element a (that is not the identity element) and \(H = \{ {a^n}|n \in {\mathbb{Z}^ + }\} \), where \({a^2} = a * a,{\text{ }}{a^3} = a * a * a\) etc.

Show that \(\{ H,{\text{ }} * \} \) is a subgroup of \(\{ G,{\text{ }} * \} \).

Write down all the elements of \(A\) and all the elements of \(B\).

Determine the symmetric difference, \(A\Delta B\), of the sets \(A\) and \(B\).

Write down all the elements of \(A \cap B,{\text{ }}A \cap C\) and \(B \cup C\).

Hence by considering \(A \cap (B \cup C)\), verify that in this case the operation \( \cap \) is distributive over the operation \( \cup \).

Show that R is an equivalence relation.

Determine the equivalence class of R containing the element \((1,{\text{ }}2)\) and illustrate this graphically.

(i)     Write down the Cayley table for \(\{ G,{\text{ }}{ \times _7}\} \) .

(ii)     Determine whether or not \(\{ G,{\text{ }}{ \times _7}\} \) is cyclic.

(iii)     Find the subgroup of G of order 3, denoting it by H .

(iv)     Identify the element of order 2 in G and find its coset with respect to H .

The group \(\{ K,{\text{ }} \circ \} \) is defined on the six permutations of the integers 1, 2, 3 and \( \circ \) denotes composition of permutations.

(i)     Show that \(\{ K,{\text{ }} \circ \} \) is non-Abelian.

(ii)     Giving a reason, state whether or not \(\{ G,{\text{ }}{ \times _7}\} \) and \(\{ K,{\text{ }} \circ \} \) are isomorphic.

Find the order of \(\{ G,{\text{ }} \circ \} \).

State the identity element in \(\{ G,{\text{ }} \circ \} \).

Find

(i)     \(p \circ p\);

(ii)     the inverse of \(p \circ p\).

(i)     Find the maximum possible order of an element in \(\{ H,{\text{ }} \circ \} \).

(ii)     Give an example of an element with this order.

Show that R is an equivalence relation.

Find the equivalence class containing 0.

Denote the equivalence class containing n by C_n .

List the first six elements of \({C_1}\).

Denote the equivalence class containing n by C_n .

Prove that \({C_n} = {C_{n + 7}}\) for all \(n \in \mathbb{N}\).

Prove that \(f\) is an injection.

(i)     Prove that \(f\) is a surjection.

(ii)     Hence, or otherwise, write down the inverse function \({f^{ - 1}}\).

Show that \(R\) is an equivalence relation.

Given that \(n = 2\) and \(p = 7\), determine the first four members of each of the four equivalence classes of \(R\).

Simplify \(\frac{c}{2} * \frac{{3c}}{4}\) .

State the identity element for G under \( * \).

For \(x \in G\) find an expression for \({x^{ - 1}}\) (the inverse of x under \( * \)).

Show that the binary operation \( * \) is commutative on G .

Show that the binary operation \( * \) is associative on G .

(i)     If \(x,{\text{ }}y \in G\) explain why \((c - x)(c - y) > 0\) .

(ii)     Hence show that \(x + y < c + \frac{{xy}}{c}\) .

Show that G is closed under \( * \).

Explain why \(\{ G, * \} \) is an Abelian group.

Below are the graphs of the two functions \(F:P \to Q{\text{ and }}g:A \to B\) .

Determine, with reference to features of the graphs, whether the functions are injective and/or surjective.

Given two functions \(h:X \to Y{\text{ and }}k:Y \to Z\) . 

Show that

(i)     if both h and k are injective then so is the composite function \(k \circ h\) ;

(ii)     if both h and k are surjective then so is the composite function \(k \circ h\) .

Find the order of elements 5, 7 and 17 in \(\{ G,{\text{ }}{ \times _{18}}\} \).

State whether or not \(\{ G,{\text{ }}{ \times _{18}}\} \) is cyclic, justifying your answer.

Write down the elements in set \(K\).

Find the left cosets of \(K\) in \(\{ G,{\text{ }}{ \times _{18}}\} \).

Prove that the function \(f\) is a homomorphism from the group \(\{ \mathbb{Z},{\text{ }} + \} {\text{ to }}\{ D,{\text{ }}{ \times _3}\} \).

Find the kernel of \(f\).

Prove that \(\{ {\text{Ker}}(f),{\text{ }} + \} \) is a subgroup of \(\{ \mathbb{Z},{\text{ }} + \} \).

Associativity and commutativity are two of the five conditions for a set S with the binary operation \( * \) to be an Abelian group; state the other three conditions.

The Cayley table for the binary operation \( \odot \) defined on the set T = {p, q, r, s, t} is given below.

(i)     Show that exactly three of the conditions for {T , \( \odot \)} to be an Abelian group are satisfied, but that neither associativity nor commutativity are satisfied. 

(ii)     Find the proper subsets of T that are groups of order 2, and comment on your result in the context of Lagrange’s theorem. 

(iii)     Find the solutions of the equation \((p \odot x) \odot x = x \odot p\) .

Show that \(\{ \mathbb{Z},{\text{ }} * \} \) is an Abelian group.

Show that there is no element of order 2.

Find a proper subgroup of \(\{ \mathbb{Z},{\text{ }} * \} \).

Show that the groups \(\{ \mathbb{Z},{\text{ }} * \} \) and \(\{ \mathbb{Z},{\text{ }} \circ \} \) are isomorphic.

Show that \(x * y \in S\) for all \(x,{\text{ }}y \in S\).

Show that the operation \( * \) on the set \(S\) is commutative.

Show that the operation \( * \) on the set \(S\) is associative.

Show that 2 is the identity element.

Show that each element \(a \in S\) has an inverse.

Given that \(R = (P \cap Q')'\) , list the elements of R .

For a set S , let \({S^ * }\) denote the set of all subsets of S ,

(i)     find \({P^ * }\) ;

(ii)     find \(n({R^ * })\) .

Show that \(R\) is an equivalence relation.

Determine the equivalence classes of \(R\).

Determine the number of equivalence classes of \(S\).

State the possible orders of an element of \(\{ G,{\text{ }} * \} \) and for each order give an example of an element of that order.

(i)     State a generator for \(\{ H,{\text{ }} * \} \).

(ii)     Write down the elements of \(\{ H,{\text{ }} * \} \).

(iii)     Write down the elements of the coset of \(H\) containing \(a\).

Show that \(R\) is reflexive.

Show that \(R\) is symmetric.

Show, by means of an example, that \(R\) is not transitive.

(i)     Show that \(gh{g^{ - 1}}\) has order 2 for all \(g \in G\).

(ii)     Deduce that gh = hg for all \(g \in G\).

Two functions, F and G , are defined on \(A = \mathbb{R}\backslash \{ 0,{\text{ }}1\} \) by

\[F(x) = \frac{1}{x},{\text{ }}G(x) = 1 - x,{\text{ for all }}x \in A.\]

(a)     Show that under the operation of composition of functions each function is its own inverse.

(b)     F and G together with four other functions form a closed set under the operation of composition of functions.

Find these four functions.

Find the element \(e\) such that \(e * y = y\), for all \(y \in S\).

(i)     Find the least solution of \(x * x = e\).

(ii)     Deduce that \((S,{\text{ }} * )\) is not a group.

Determine whether or not \(e\) is an identity element.

Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow x + y > 7\) is

(i)     reflexive;

(ii)     symmetric;

(iii)     transitive.

Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow - 2 < x - y < 2\) is

(i)     reflexive;

(ii)     symmetric;

(iii)     transitive.

Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow xy > 0\) is

(i)     reflexive;

(ii)     symmetric;

(iii)     transitive.

Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow \frac{x}{y} \in \mathbb{Z}\) is

(i)     reflexive;

(ii)     symmetric;

(iii)     transitive.

One of the relations from parts (a), (b), (c) and (d) is an equivalence relation.

For this relation, state what the equivalence classes are.

Write down all four subsets of A .

Construct the Cayley table for \(P(A)\) under \(\Delta \) .

Prove that \(\left\{ {P(A),{\text{ }}\Delta } \right\}\) is a group. You are allowed to assume that \(\Delta \) is associative.

Is \(\{ P(A){\text{, }}\Delta \} \) isomorphic to \(\{ {\mathbb{Z}_4},{\text{ }}{ + _4}\} \) ? Justify your answer.

(i)     State the identity element for \(\{ P(S){\text{, }}\Delta \} \).

(ii)     Write down \({X^{ - 1}}\) for \(X \in P(S)\) .

(iii)     Hence prove that \(\{ P(S){\text{, }}\Delta \} \) is a group.

Explain why \(\{ P(S){\text{, }} \cup \} \) is not a group.

Explain why \(\{ P(S){\text{, }} \cap \} \) is not a group.

Copy and complete the following Cayley table for \(\{ T,{\text{ }} * \} \).

Prove that \(\{ T,{\text{ }} * \} \) forms an Abelian group.

Find the order of each element in \(T\).

Given that \(\{ H,{\text{ }} * \} \) is the subgroup of \(\{ T,{\text{ }} * \} \) of order \(2\), partition \(T\) into the left cosets with respect to \(H\).

Show that \(f\) is a bijection.

Hence write down the inverse function \({f^{ - 1}}(x,{\text{ }}y)\).

(i)     Sketch the graph of \(y = f(x)\) and hence justify whether or not \(f\) is a bijection.

(ii)     Show that \(A\) is a group under the binary operation of multiplication.

(iii)     Give a reason why \(B\) is not a group under the binary operation of multiplication.

(iv)     Find an example to show that \(f(a \times b) = f(a) \times f(b)\) is not satisfied for all \(a,{\text{ }}b \in A\).

(i)     Sketch the graph of \(y = g(x)\) and hence justify whether or not \(g\) is a bijection.

(ii)     Show that \(g(a + b) = g(a) \times g(b)\) for all \(a,{\text{ }}b \in \mathbb{R}\).

(iii)     Given that \(\{ \mathbb{R},{\text{ }} + \} \) and \(\{ D,{\text{ }} \times \} \) are both groups, explain whether or not they are isomorphic.

(a)     Show that \(f:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) defined by \(f(x,{\text{ }}y) = (2x + y,{\text{ }}x - y)\) is a bijection.

(b)     Find the inverse of f .

(i)     Show that \( * \) is commutative.

(ii)     Find the identity element.

(iii)     Find the inverse of the element a .

The binary operation \( \cdot \) is defined on \(\mathbb{R}\) as follows. For any elements a , \(b \in \mathbb{R}\)

\(a \cdot b = 3ab\) . The set S is the set of all ordered pairs \((x,{\text{ }}y)\) of real numbers and the binary operation \( \odot \) is defined on the set S as

\(({x_1},{\text{ }}{y_1}) \odot ({x_2},{\text{ }}{y_2}) = ({x_1} * {x_2},{\text{ }}{y_1} \cdot {y_2}){\text{ }}.\)

Determine whether or not \( \odot \) is associative.

(a)     Draw the Cayley table for the set of integers G = {0, 1, 2, 3, 4, 5} under addition modulo 6, \({ + _6}\).

(b)     Show that \(\{ G,{\text{ }}{ + _6}\} \) is a group.

(c)     Find the order of each element.

(d)     Show that \(\{ G,{\text{ }}{ + _6}\} \) is cyclic and state its generators.

(e)     Find a subgroup with three elements. 

(f)     Find the other proper subgroups of \(\{ G,{\text{ }}{ + _6}\} \).

The function \(f:[0,{\text{ }}\infty [ \to [0,{\text{ }}\infty [\) is defined by \(f(x) = 2{{\text{e}}^x} + {{\text{e}}^{ - x}} - 3\) .

(a)     Find \(f'(x)\) .

(b)     Show that f is a bijection.

(c)     Find an expression for \({f^{ - 1}}(x)\) .

A \ B ;

\(A\Delta B\) .

is closed;

is commutative;

is associative;

has an identity element.

Find the values represented by each of the letters in the table.

Find the order of each of the elements of the group.

Write down the three sets that form subgroups of order 2.

Find the three sets that form subgroups of order 4.

Given that p , q and r are elements of a group, prove the left-cancellation rule, i.e. \(pq = pr \Rightarrow q = r\) .

Your solution should indicate which group axiom is used at each stage of the proof.

Consider the group G , of order 4, which has distinct elements a , b and c and the identity element e .

(i)     Giving a reason in each case, explain why ab cannot equal a or b .

(ii)     Given that c is self inverse, determine the two possible Cayley tables for G .

(iii)     Determine which one of the groups defined by your two Cayley tables is isomorphic to the group defined by the set {1, −1, i, −i} under multiplication of complex numbers. Your solution should include a correspondence between a, b, c, e and 1, −1, i, −i .

A binary operation is defined on {−1, 0, 1} by

\[A \odot B = \left\{ {\begin{array}{*{20}{c}}
  { - 1,}&{{\text{if }}\left| A \right| < \left| B \right|} \\
  {0,}&{{\text{if }}\left| A \right| = \left| B \right|} \\
  {1,}&{{\text{if }}\left| A \right| > \left| B \right|{\text{.}}}
\end{array}} \right.\]

(a)     Construct the Cayley table for this operation.

(b)     Giving reasons, determine whether the operation is

(i)     closed;

(ii)     commutative;

(iii)     associative.

(i)     Sketch the set \(X \times Y\) in the Cartesian plane.

(ii)     Sketch the set \(Y \times X\) in the Cartesian plane.

(iii)     State \((X \times Y) \cap (Y \times X)\).

Consider the function \(f:X \times Y \to \mathbb{R}\) defined by \(f(x,{\text{ }}y) = x + y\) and the function \(g:X \times Y \to \mathbb{R}\) defined by \(g(x,{\text{ }}y) = xy\).

(i)     Find the range of the function f.

(ii)     Find the range of the function g.

(iii)     Show that \(f\) is an injection.

(iv)     Find \({f^{ - 1}}(\pi )\), expressing your answer in exact form.

(v)     Find all solutions to \(g(x,{\text{ }}y) = \frac{1}{2}\).

Prove that \(f({e_G}) = {e_H}\), where \({e_G}\) is the identity element in \(G\) and \({e_H}\) is the identity

element in \(H\).

(i)     Prove that the kernel of \(f,{\text{ }}K = {\text{Ker}}(f)\), is closed under the group operation.

(ii)     Deduce that \(K\) is a subgroup of \(G\).

(i)     Prove that \(gk{g^{ - 1}} \in K\) for all \(g \in G,{\text{ }}k \in K\).

(ii)     Deduce that each left coset of K in G is also a right coset.

Prove that: 

(i)     \(f\) is an injection,

(ii)     \(g\) is a surjection.

Given that \(X = {\mathbb{R}^ + } \cup \{ 0\} \) and \(Y = \mathbb{R}\), choose a suitable pair of functions \(f\) and \(g\) to show that \(g\) is not necessarily a bijection.

Let \((H,{\text{ }} * {\text{)}}\) be a subgroup of the group \((G,{\text{ }} * {\text{)}}\).

Consider the relation \(R\) defined in \(G\) by \(xRy\) if and only if \({y^{ - 1}} * x \in H\).

(a)     Show that \(R\) is an equivalence relation on \(G\).

(b)     Determine the equivalence class containing the identity element.

Two members of \(A\) are given by \(p = (1{\text{ }}2{\text{ }}5)\) and \(q = (1{\text{ }}3)(2{\text{ }}5)\).

Find the single permutation which is equivalent to \(q \circ p\).

State a permutation belonging to \(A\) of order

(i)     \(4\);

(ii)     \(6\).

Let \(P = \) {all permutations in \(A\) where exactly two integers change position},

and \(Q = \) {all permutations in \(A\) where the integer \(1\) changes position}.

(i)     List all the elements in \(P \cap Q\).

(ii)     Find \(n(P \cap Q')\).

Given the sets \(A\) and \(B\), use the properties of sets to prove that \(A \cup (B' \cup A)' = A \cup B\), justifying each step of the proof.

(a)     Write down why the table below is a Latin square.

\[\begin{gathered}
  \begin{array}{*{20}{c}}
  {}&d&e&b&a&c
\end{array} \\
  \begin{array}{*{20}{c}}
  d \\
  e \\
  b \\
  a \\
  c
\end{array}\left[ {\begin{array}{*{20}{c}}
  c&d&e&b&a \\
  d&e&b&a&c \\
  a&b&d&c&e \\
  b&a&c&e&d \\
  e&c&a&d&b
\end{array}} \right] \\
\end{gathered} \]

(b)     Use Lagrange’s theorem to show that the table is not a group table.

Let \(p = {2^k} + 1,{\text{ }}k \in {\mathbb{Z}^ + }\) be a prime number and let G be the group of integers 1, 2, ..., p − 1 under multiplication defined modulo p.

By first considering the elements \({2^1},{\text{ }}{2^2},{\text{ ..., }}{2^k}\) and then the elements \({2^{k + 1}},{\text{ }}{2^{k + 2}},{\text{ …,}}\) show that the order of the element 2 is 2k.

Deduce that \(k = {2^n}{\text{ for }}n \in \mathbb{N}\) .

Prove that \((A \cap B)\backslash (A \cap C) = A \cap (B\backslash C)\) where A, B and C are three subsets of the universal set U.

Let \(\{ G,{\text{ }} * \} \) be a finite group and let H be a non-empty subset of G . Prove that \(\{ H,{\text{ }} * \} \) is a group if H is closed under \( * \).

Prove that for all \(a \in G,{\text{ }}f({a^{ - 1}}) = {\left( {f(a)} \right)^{ - 1}}\).

Let \(\{ H,{\text{ }} \circ \} \) be the cyclic group of order seven, and let \(p\) be a generator.

Let \(x \in G\) such that \(f(x) = {p^{\text{2}}}\).

Find \(f({x^{ - 1}})\).

Given that \(f(x * y) = p\), find \(f(y)\).

H and K are subgroups of a group G. By considering the four group axioms, prove that \(H \cap K\) is also a subgroup of G.

Prove that set difference is not associative.

Prove that \(f\) is an injection.

Prove that \(f\) is not a surjection.

Show that \((G,{\text{ }} + )\) forms a group where \( + \) denotes addition on \(\mathbb{Q}\). Associativity may be assumed.

Assuming that \((H,{\text{ }} + )\) forms a group, show that it is a proper subgroup of \((G,{\text{ }} + )\).

The mapping \(\phi :G \to G\) is given by \(\phi (g) = g + g\), for \(g \in G\).

Prove that \(\phi \) is an isomorphism.

Consider the following functions

     \(f:\left] {1,{\text{ }} + \infty } \right[ \to {\mathbb{R}^ + }\) where \(f(x) = (x - 1)(x + 2)\)

     \(g:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) where \(g(x,{\text{ }}y) = \left( {\sin (x + y),{\text{ }}x + y} \right)\)

     \(h:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\) where \(h(x,{\text{ }}y) = (x + 3y,{\text{ }}2x + y)\)

(a)     Show that \(f\) is bijective.

(b)     Determine, with reasons, whether

(i)     \(g\) is injective;

(ii)     \(g\) is surjective.

(c)     Find an expression for \({h^{ - 1}}(x,{\text{ }}y)\) and hence justify that \(h\) has an inverse function.

Let \(f:\mathbb{Z} \times \mathbb{R} \to \mathbb{R},{\text{ }}f(m,{\text{ }}x) = {( - 1)^m}x\). Determine whether f is

(i)     surjective;

(ii)     injective.

P is the set of all polynomials such that \(P = \left\{ {\sum\limits_{i = 0}^n {{a_i}{x^i}|n \in \mathbb{N}} } \right\}\).

Let \(g:P \to P,{\text{ }}g(p) = xp\). Determine whether g is

(i)     surjective;

(ii)     injective.

Let \(h:\mathbb{Z} \to {\mathbb{Z}^ + }\), \(h(x) = \left\{ {\begin{array}{*{20}{c}}
  {2x,}&{x > 0} \\
  {1 - 2x,}&{x \leqslant 0}
\end{array}} \right\}\). Determine whether h is

(i)     surjective;

(ii)     injective.

Prove that \(f\) is

(i)     not injective;

(ii)     not surjective.

The relation \(R\) is defined for \(a,{\text{ }}b \in \mathbb{R}\) so that \(aRb\) if and only if \(f(a) \times f(b) = 1\).

Show that \(R\) is an equivalence relation.

The relation \(R\) is defined for \(a,{\text{ }}b \in \mathbb{R}\) so that \(aRb\) if and only if \(f(a) \times f(b) = 1\).

State the equivalence classes of \(R\).

The function \(f:{\mathbb{R}^ + } \times {\mathbb{R}^ + } \to {\mathbb{R}^ + } \times {\mathbb{R}^ + }\) is defined by \(f(x,{\text{ }}y) = \left( {x{y^2},\frac{x}{y}} \right)\).

Show that f is a bijection.

Let G be a finite cyclic group.

(a)     Prove that G is Abelian.

(b)     Given that a is a generator of G, show that \({a^{ - 1}}\) is also a generator.

(c)     Show that if the order of G is five, then all elements of G, apart from the identity, are generators of G.

The function \(f:\mathbb{R} \to \mathbb{R}\) is defined by

\[f(x) = 2{{\text{e}}^x} - {{\text{e}}^{ - x}}.\]

(a)     Show that f is a bijection.

(b)     Find an expression for \({f^{ - 1}}(x)\).

Determine the order of S₄.

Find the proper subgroup H of order 6 containing \({p_1}\), \({p_2}\) and their compositions. Express each element of H in cycle form.

Let \(f{\text{:}}\,{S_4} \to {S_4}\) be defined by \(f\left( p \right) = p \circ p\) for \(p \in {S_4}\).

Using \({p_1}\) and \({p_2}\), explain why \(f\) is not a homomorphism.

The relation aRb is defined on {1, 2, 3, 4, 5, 6, 7, 8, 9} if and only if ab is the square of a positive integer. 

(i)     Show that R is an equivalence relation. 

(ii)     Find the equivalence classes of R that contain more than one element.

Given the group \((G,{\text{ }} * )\), a subgroup \((H,{\text{ }} * )\) and \(a,{\text{ }}b \in G\), we define \(a \sim b\) if and only if \(a{b^{ - 1}} \in H\). Show that \( \sim \) is an equivalence relation.

Set \(S = \{ {x_0},{\text{ }}{x_1},{\text{ }}{x_2},{\text{ }}{x_3},{\text{ }}{x_4},{\text{ }}{x_5}\} \) and a binary operation \( \circ \) on S is defined as \({x_i} \circ {x_j} = {x_k}\), where \(i + j \equiv k(\bmod 6)\).

(a)     (i)     Construct the Cayley table for \(\{ S,{\text{ }} \circ \} \) and hence show that it is a group.

  (ii)     Show that \(\{ S,{\text{ }} \circ \} \) is cyclic.

(b)     Let \(\{ G,{\text{ }} * \} \) be an Abelian group of order 6. The element \(a \in {\text{G}}\) has order 2 and the element \(b \in {\text{G}}\) has order 3.

  (i)     Write down the six elements of \(\{ G,{\text{ }} * \} \).

  (ii)     Find the order of \({\text{a}} * b\) and hence show that \(\{ G,{\text{ }} * \} \) is isomorphic to \(\{ S,{\text{ }} \circ \} \).

(a)     Find the range of f .

(b)     Prove that f is an injection.

(c)     Taking the codomain of f to be equal to the range of f , find an expression for \({f^{ - 1}}(x)\) .

The relation R is defined on \(\mathbb{Z} \times \mathbb{Z}\) such that \((a,{\text{ }}b)R(c,{\text{ }}d)\) if and only if a − c is divisible by 3 and b − d is divisible by 2.

(a)     Prove that R is an equivalence relation.

(b)     Find the equivalence class for (2, 1) .

(c)     Write down the five remaining equivalence classes.

The binary operation \( * \) is defined on the set S = {0, 1, 2, 3} by

\[a * b = a + 2b + ab(\bmod 4){\text{ .}}\]

(a)     (i)     Construct the Cayley table.

  (ii)     Write down, with a reason, whether or not your table is a Latin square.

(b)     (i)     Write down, with a reason, whether or not \( * \) is commutative.

  (ii)     Determine whether or not \( * \) is associative, justifying your answer.

(c)     Find all solutions to the equation \(x * 1 = 2 * x\) , for \(x \in S\) .

(a)     Find the six roots of the equation \({z^6} - 1 = 0\) , giving your answers in the form \(r\,{\text{cis}}\,\theta {\text{, }}r \in {\mathbb{R}^ + }{\text{, }}0 \leqslant \theta  < 2\pi \) .

(b)     (i)     Show that these six roots form a group G under multiplication of complex numbers.

          (ii)     Show that G is cyclic and find all the generators.

          (iii)     Give an example of another group that is isomorphic to G, stating clearly the corresponding elements in the two groups.

The relation R is defined on \({\mathbb{Z}^ + }\) by aRb if and only if ab is even. Show that only one of the conditions for R to be an equivalence relation is satisfied.

The relation S is defined on \({\mathbb{Z}^ + }\) by aSb if and only if \({a^2} \equiv {b^2}(\bmod 6)\) .

(i)     Show that S is an equivalence relation.

(ii)     For each equivalence class, give the four smallest members.

The groups \(\{ K,{\text{ }} * \} \) and \(\{ H,{\text{ }} \odot \} \) are defined by the following Cayley tables.

G    

H    

By considering a suitable function from G to H , show that a surjective homomorphism exists between these two groups. State the kernel of this homomorphism.

Three functions mapping \(\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}\) are defined by

\[{f_1}(m,{\text{ }}n) = m - n + 4;\,\,\,{f_2}(m,{\text{ }}n) = \left| m \right|;\,\,\,{f_3}(m,{\text{ }}n) = {m^2} - {n^2}.\]

Two functions mapping \(\mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}\) are defined by

\[{g_1}(k) = (2k,{\text{ }}k);\,\,\,{g_2}(k) = \left( {k,{\text{ }}\left| k \right|} \right).\]

(a)     Find the range of

(i)     \({f_1} \circ {g_1}\) ;

(ii)     \({f_3} \circ {g_2}\) .

(b)     Find all the solutions of \({f_1} \circ {g_2}(k) = {f_2} \circ {g_1}(k)\) .

(c)     Find all the solutions of \({f_3}(m,{\text{ }}n) = p\) in each of the cases p =1 and p = 2 .

State Lagrange’s theorem.

Verify that the inverse of \(a * {b^{ - 1}}\) is equal to \(b * {a^{ - 1}}\).

Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\). Let \(R\) be a relation defined on \(G\) by

\[aRb \Leftrightarrow a * {b^{ - 1}} \in H.\]

Prove that \(R\) is an equivalence relation, indicating clearly whenever you are using one of the four properties required of a group.

Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\) .Let \(R\) be a relation defined on \(G\) by

\[aRb \Leftrightarrow a * {b^{ - 1}} \in H.\]

Show that \(aRb \Leftrightarrow a \in Hb\), where \(Hb\) is the right coset of \(H\) containing \(b\).

Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\) .Let \(R\) be a relation defined on \(G\) by

\[aRb \Leftrightarrow a * {b^{ - 1}} \in H.\]

It is given that the number of elements in any right coset of \(H\) is equal to the order of \(H\).

Explain how this fact together with parts (c) and (d) prove Lagrange’s theorem.

(a)     Given a set \(U\), and two of its subsets \(A\) and \(B\), prove that

\[(A\backslash B) \cup (B\backslash A) = (A \cup B)\backslash (A \cap B),{\text{ where }}A\backslash B = A \cap B'.\]

(b)     Let \(S = \{ A,{\text{ }}B,{\text{ }}C,{\text{ }}D\} \) where \(A = \emptyset ,{\text{ }}B = \{ 0\} ,{\text{ }}C = \{ 0,{\text{ }}1\} \) and \(D = \{ {\text{0, 1, 2}}\} \).

State, with reasons, whether or not each of the following statements is true.

(i)     The operation \ is closed in \(S\).

(ii)     The operation \( \cap \) has an identity element in \(S\) but not all elements have an inverse.

(iii)     Given \(Y \in S\), the equation \(X \cup Y = Y\) always has a unique solution for \(X\) in \(S\).

Show that the product of three consecutive integers is divisible by \(6\).

Hence prove that \(R\) is reflexive.

Find the set of all \(y\) for which \(5Ry\).

Find the set of all \(y\) for which \(3Ry\).

Using your answers for (c) and (d) show that \(R\) is not symmetric.

Determine, giving reasons, which of the following sets form groups under the operations given below. Where appropriate you may assume that multiplication is associative.

(a)     \(\mathbb{Z}\) under subtraction.

(b)     The set of complex numbers of modulus 1 under multiplication.

(c)     The set {1, 2, 4, 6, 8} under multiplication modulo 10.

(d)     The set of rational numbers of the form

\[\frac{{3m + 1}}{{3n + 1}},{\text{ where }}m,{\text{ }}n \in \mathbb{Z}\]

under multiplication.

(i)     Write down the six smallest non-negative elements of \(S\).

(ii)     Show that \(\{ S,{\text{ }} + \} \) is a group.

(iii)     Give a reason why \(\{ S,{\text{ }} \times \} \) is not a group. Justify your answer.

The relation \(R\) is defined on \(S\) by \({s_1}R{s_2}\) if \(3{s_1} + 5{s_2} \in \mathbb{Z}\).

(i)     Show that \(R\) is an equivalence relation.

(ii)     Determine the equivalence classes.

The binary operation \(\Delta\) is defined on the set \(S =\) {1, 2, 3, 4, 5} by the following Cayley table.

(a)     State whether S is closed under the operation Δ and justify your answer.

(b)     State whether Δ is commutative and justify your answer.

(c)     State whether there is an identity element and justify your answer.

(d)     Determine whether Δ is associative and justify your answer.

(e)     Find the solutions of the equation \(a\Delta b = 4\Delta b\), for \(a \ne 4\).

(a)     Determine whether or not \( * \) is

(i)     closed,

(ii)     commutative,

(iii)     associative.

(b)     (i)     Find the identity element.

(ii)     Find the set of positive integers having an inverse under \( * \).

\(A\), \(B\), \(C\) and \(D\) are subsets of \(\mathbb{Z}\) .

\(A = \{ \left. m \right|m{\text{ is a prime number less than 15}}\}\)

\(B = \{ \left. m \right|{m^4} = 8m\} \)

\(C = \{ \left. m \right|(m + 1)(m - 2) < 0\} \)

\(D = \{ \left. m \right|{m^2} < 2m + 4\} \)

(a)     List the elements of each of these sets.

(b)     Determine, giving reasons, which of the following statements are true and which are false.

  (i)     \(n(D) = n(B) + n(B \cup C)\)

  (ii)     \(D\backslash B \subset A\)

  (iii)     \(B \cap A' = \emptyset \)

  (iv)     \(n(B\Delta C) = 2\)

(a)     Consider the set A = {1, 3, 5, 7} under the binary operation \( * \), where \( * \) denotes multiplication modulo 8.

  (i)     Write down the Cayley table for \(\{ A,{\text{ }} * \} \).

  (ii)     Show that \(\{ A,{\text{ }} * \} \) is a group.

  (iii)     Find all solutions to the equation \(3 * x * 7 = y\). Give your answers in the form \((x,{\text{ }}y)\).

(b)     Now consider the set B = {1, 3, 5, 7, 9} under the binary operation \( \otimes \), where \( \otimes \) denotes multiplication modulo 10. Show that \(\{ B,{\text{ }} \otimes \} \) is not a group.

(c)     Another set C can be formed by removing an element from B so that \(\{ C,{\text{ }} \otimes \} \) is a group.

  (i)     State which element has to be removed.

  (ii)     Determine whether or not \(\{ A,{\text{ }} * \} \) and \(\{ C,{\text{ }} \otimes \} \) are isomorphic.

Let {G , \( * \)} be a finite group of order n and let H be a non-empty subset of G .

(a)     Show that any element \(h \in H\) has order smaller than or equal to n .

(b)     If H is closed under \( * \), show that {H , \( * \)} is a subgroup of {G , \( * \)}.

The group \(\{ G,{\text{ }} * \} \) is Abelian and the bijection \(f:{\text{ }}G \to G\) is defined by \(f(x) = {x^{ - 1}},{\text{ }}x \in G\).

Show that \(f\) is an isomorphism.

Show that R is an equivalence relation.

The Cayley table for G is shown below.

The subgroup H is given as \(H = \{ e,{\text{ }}{a^2}b\} \).

(i)     Find the equivalence class with respect to R which contains ab.

(ii)     Another equivalence relation \(\rho \) is defined on G by \(x\rho y\) if and only if \({x^{ - 1}}y \in H\), for \(x,{\text{ }}y \in G\). Find the equivalence class with respect to \(\rho \) which contains ab.

Show that R is an equivalence relation.

Show that the equivalence defining R can be written in the form

\[(a - b)(a + b + 1) \equiv 0(\bmod 5).\]

Hence, or otherwise, determine the equivalence classes.

Complete the following Cayley table

[5 marks]

(i)     State the inverse of each element.

(ii)     Determine the order of each element.

Write down the subgroups containing

(i)     \(r\),

(ii)     \(u\).

The permutation \({p_1}\) of the set {1, 2, 3, 4} is defined by

\[{p_1} = \left( {\begin{array}{*{20}{c}}
  1&2&3&4 \\
  2&4&1&3
\end{array}} \right)\]

(a)     (i)     State the inverse of \({p_1}\).

  (ii)     Find the order of \({p_1}\).

(b)     Another permutation \({p_2}\) is defined by

\[{p_2} = \left( {\begin{array}{*{20}{c}}
  1&2&3&4 \\
  3&2&4&1
\end{array}} \right)\]

  (i)     Determine whether or not the composition of \({p_1}\) and \({p_2}\) is commutative.

  (ii)     Find the permutation \({p_3}\) which satisfies

\[{p_1}{p_3}{p_2} = \left( {\begin{array}{*{20}{c}}
  1&2&3&4 \\
  1&2&3&4
\end{array}} \right){\text{.}}\]

Let R be a relation on the set \(\mathbb{Z}\) such that \(aRb \Leftrightarrow ab \geqslant 0\), for a, b \( \in \mathbb{Z}\).

(a)     Determine whether R is

(i)     reflexive;

(ii)     symmetric;

(iii)     transitive.

(b)     Write down with a reason whether or not R is an equivalence relation.

Show that R is an equivalence relation.

Identify the three equivalence classes.

(i)      Prove that \(G\) is cyclic and state two of its generators.

(ii)     Let \(H\) be the subgroup generated by \({a^4}\). Construct a Cayley table for \(H\).

State, with a reason, whether or not it is necessary that a group is cyclic given that all its proper subgroups are cyclic.

(a)     Show that {1, −1, i, −i} forms a group of complex numbers G under multiplication.

(b)     Consider \(S = \{ e,{\text{ }}a,{\text{ }}b,{\text{ }}a * b\} \) under an associative operation \( * \) where e is the identity element. If \(a * a = b * b = e\) and \(a * b = b * a\) , show that

(i)     \(a * b * a = b\) ,

(ii)     \(a * b * a * b = e\) .

(c)     (i)     Write down the Cayley table for \(H = \{ S{\text{ , }} * \} \).

(ii)     Show that H is a group.

(iii)     Show that H is an Abelian group.

(d)     For the above groups, G and H , show that one is cyclic and write down why the other is not. Write down all the generators of the cyclic group.

(e)     Give a reason why G and H are not isomorphic.

The relations R and S are defined on quadratic polynomials P of the form

\[P(z) = {z^2} + az + b{\text{ , where }}a{\text{ , }}b \in \mathbb{R}{\text{ , }}z \in \mathbb{C}{\text{ .}}\]

(a)     The relation R is defined by \({P_1}R{P_2}\) if and only if the sum of the two zeros of \({P_1}\) is equal to the sum of the two zeros of \({P_2}\) .

(i)     Show that R is an equivalence relation.

(ii)     Determine the equivalence class containing \({z^2} - 4z + 5\) .

(b)     The relation S is defined by \({P_1}S{P_2}\) if and only if \({P_1}\) and \({P_2}\) have at least one zero in common. Determine whether or not S is transitive.

(a)     Show that R is an equivalence relation.

(b)     Describe, geometrically, the equivalence classes.

Copy and complete the following Cayley table for this binary operation.

Give one reason why \(\{ S,{\text{ }}{ \times _{14}}\} \) is not a group.

Show that a new set G can be formed by removing one of the elements of S such that \(\{ G,{\text{ }}{ \times _{14}}\} \) is a group.

Determine the order of each element of \(\{ G,{\text{ }}{ \times _{14}}\} \).

Find the proper subgroups of \(\{ G,{\text{ }}{ \times _{14}}\} \).

Copy and complete the following operation table.

(i)     Show that {S , \( * \)} is a group.

(ii)     Find the order of each element of {S , \( * \)}.

(iii)     Hence show that {S , \( * \)} is cyclic and find all the generators.

The set T is defined by \(\{ x * x:x \in S\} \). Show that {T , \( * \)} is a subgroup of {S , \( * \)}.

The group \(\{ G,{\text{ }} * \} \) is defined on the set \(G\) with binary operation \( * \). \(H\) is a subset of \(G\) defined by \(H = \{ x:{\text{ }}x \in G,{\text{ }}a * x * {a^{ - 1}} = x{\text{ for all }}a \in G\} \). Prove that \(\{ H,{\text{ }} * \} \) is a subgroup of \(\{ G,{\text{ }} * \} \).

Copy and complete the table.

Show that \(\{ S,{\text{ }} * \} \) is an Abelian group.

Determine the orders of all the elements of \(\{ S,{\text{ }} * \} \).

(i)     Find the two proper subgroups of \(\{ S,{\text{ }} * \} \).

(ii)     Find the coset of each of these subgroups with respect to the element 5.

Solve the equation \(2 * x * 4 * x * 4 = 2\).

Show that {T, ×₁₀} is a group. (You may assume associativity.)

By making reference to the Cayley table, explain why T is Abelian.

Find the order of each element of {T, ×₁₀}.

Hence show that {T, ×₁₀} is cyclic and write down all its generators.

The binary operation multiplication modulo 10, denoted by ×₁₀ , is defined on the set V = {1, 3 ,5 ,7 ,9}.

Show that {V, ×₁₀} is not a group.

Find \(\left( {A \cup B} \right) \cap \left( {A \cup C} \right)\).

Verify that A \ C ≠ C \ A.

Let S be a set containing \(n\) elements where \(n \in \mathbb{N}\).

Show that S has \({2^n}\) subsets.

Consider the following Cayley table for the set G = {1, 3, 5, 7, 9, 11, 13, 15} under the operation \({ \times _{16}}\), where \({ \times _{16}}\) denotes multiplication modulo 16.

(i)     Find the values of a, b, c, d, e, f, g, h, i and j.

(ii)     Given that \({ \times _{16}}\) is associative, show that the set G, together with the operation \({ \times _{16}}\), forms a group.

The Cayley table for the set \(H = \{ e,{\text{ }}{a_1},{\text{ }}{a_2},{\text{ }}{a_3},{\text{ }}{b_1},{\text{ }}{b_2},{\text{ }}{b_3},{\text{ }}{b_4}\} \) under the operation \( * \), is shown below.

(i)     Given that \( * \) is associative, show that H together with the operation \( * \) forms a group.

(ii)     Find two subgroups of order 4.

Show that \(\{ G,{\text{ }}{ \times _{16}}\} \) and \(\{ H,{\text{ }} * \} \) are not isomorphic.

Show that \(\{ H,{\text{ }} * \} \) is not cyclic.

The function \(g:\mathbb{Z} \to \mathbb{Z}\) is defined by \(g(n) = \left| n \right| - 1{\text{ for }}n \in \mathbb{Z}\) . Show that g is neither surjective nor injective.

The set S is finite. If the function \(f:S \to S\) is injective, show that f is surjective.

Using the set \({\mathbb{Z}^ + }\) as both domain and codomain, give an example of an injective function that is not surjective.

Show that if both f and g are injective, then \(g \circ f\) is also injective.

Show that if both f and g are surjective, then \(g \circ f\) is also surjective.

Show, using a single counter example, that both of the converses to the results in part (a) and part (b) are false.

(i)     Sketch the graph of f.

(ii)     By referring to your graph, show that f is a bijection.

Find \({f^{ - 1}}(x)\).

Determine, using Venn diagrams, whether the following statements are true.

(i)     \(A' \cup B' = (A \cup B)'\)

(ii)     \((A\backslash B) \cup (B\backslash A) = (A \cup B)\backslash (A \cap B)\)

Prove, without using a Venn diagram, that \(A\backslash B\) and \(B\backslash A\) are disjoint sets.