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href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../537/algebra.html">Algebra</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Deductive Proofs</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/algebra/deductive-proofs/main-1.png" style="float: left; width: 100px; height: 100px;">On this page, we will look at deductive reasoning in order to be able to make direct proofs. This is a hugely important topic in mathematics, since we like to be absolutely sure of the results we have found. However, this can be a challenging - when a problem is unfamiliar, it is often difficult to know where to start. The key is to be able to use our existing knowledge and theorems, make connections and work towards a conclusion. You might want to think of these as puzzles to solve.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn to</p> <ul> <li>Make simple algebraic deductive proofs</li> <li>Use the symbol <span class="math-tex">\(\equiv\)</span> for identities</li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/algebra/deductive-proofs/revision-notes_deductive_proofs.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/algebra/deductive-proofs/revision-notes_deductive_proofs.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#10ae62a1"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="10ae62a1"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Deductive Proof <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-452-1733" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Complete the following to<em><strong> "prove that 2 consecutive integers add together to make an odd integer"</strong></em></p><p> <span class="q-text-draggable draggable" draggable="true">n</span> <span class="q-text-draggable draggable" draggable="true">2n + 1</span> <span class="q-text-draggable draggable" draggable="true">2n</span> <span class="q-text-draggable draggable" draggable="true">n + 1</span></p></div><div class="q-answer"><p>Let <strong><em>n</em></strong> be the first integer</p><p>The second integer is <input type="text" style="height: auto;" data-c="n + 1"> <span class="review"></span></p><p>Adding them together gives <strong><em>n</em></strong> + <input type="text" style="height: auto;" data-c="n + 1"> <span class="review"></span> <span class="math-tex">\(\equiv\)</span> <input type="text" style="height: auto;" data-c="2n + 1"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(n + (n + 1) \equiv2n+1\)</span></p><p><strong><em>2n</em></strong> is an even number, so <strong><em>2n + 1 </em></strong>is odd</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <strong><em>a</em></strong> and <strong><em>b</em></strong> to prove that x² - 4x + 7 > 0</p><p>We can complete the square:</p><p>x² - 4x + 7 <span class="math-tex">\(\equiv\)</span> (x - <strong><em>a</em></strong>)² + <strong><em>b</em></strong></p><p>A square number is always greater than or equal to zero, so (x - <strong><em>a</em></strong>)² ≥ 0</p><p>Since, <strong><em>b </em></strong>≥ 0,</p><p>(x - <strong><em>a</em></strong>)² + <strong><em>b </em></strong>≥ 0</p><p>Hence x² - 4x + 7 > 0</p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>This is a good technique for showing that a function is not negative</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Complete the following to <strong><em>"prove that adding 2 consecutive odd numbers gives a multiple of 4"</em></strong></p><p> <span class="q-text-draggable draggable" draggable="true">2n</span> <span class="q-text-draggable draggable" draggable="true">4n</span> <span class="q-text-draggable draggable" draggable="true">2n + 1</span> <span class="q-text-draggable draggable" draggable="true">4n - 1</span></p></div><div class="q-answer"><p>Let <strong><em>2n - 1 </em></strong>be the first odd integer</p><p>Then <input type="text" style="height: auto;" data-c="2n + 1"> <span class="review"></span> is the next odd integer</p><p>Add them together: <strong><em>2n - 1 + </em></strong><input type="text" style="height: auto;" data-c="2n + 1"> <span class="review"></span> <span class="math-tex">\(\equiv\)</span> <input type="text" style="height: auto;" data-c="4n"> <span class="review"></span></p></div><div class="q-explanation"><p><strong><em>4n</em></strong> is divisible by 4, hence is a multiple of 4</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Complete the following to <strong><em>"prove that if m and n are either both even or both odd, then m + n is even"</em></strong></p><p> <span class="q-text-draggable draggable" draggable="true">k</span> <span class="q-text-draggable draggable" draggable="true">2k - 1</span> <span class="q-text-draggable draggable" draggable="true">j + k</span> <span class="q-text-draggable draggable" draggable="true">j + k - 1</span> <span class="q-text-draggable draggable" draggable="true">2k</span></p></div><div class="q-answer"><p>If <strong><em>m</em></strong> and <strong><em>n</em></strong> are even, then there exists integers <strong><em>j</em></strong> and <strong><em>k</em></strong> such that</p><p><strong><em>m = 2j</em></strong> and <strong><em>n = 2k</em></strong></p><p><strong><em>m + n <span class="math-tex">\(\equiv\)</span> 2j</em></strong> <strong><em>+ </em></strong><input type="text" style="height: auto;" data-c="2k"> <span class="review"></span></p><p><strong><em>m + n <span class="math-tex">\(\equiv\)</span> 2(</em></strong> <input type="text" style="height: auto;" data-c="j + k"> <span class="review"></span> <strong><em>)</em></strong></p><hr class="hidden-separator"><p>If <strong><em>m</em></strong> and <strong><em>n</em></strong> are odd, then there exists integers <strong><em>j</em></strong> and <strong><em>k</em></strong> such that</p><p><strong><em>m = 2j - 1</em></strong> and <strong><em>n = </em></strong> <input type="text" style="height: auto;" data-c="2k - 1"> <span class="review"></span></p><p><strong><em>m + n <span class="math-tex">\(\equiv\)</span> 2j</em></strong> <strong><em>- 1 + </em></strong><input type="text" style="height: auto;" data-c="2k - 1"> <span class="review"></span></p><p><strong><em>m + n <span class="math-tex">\(\equiv\)</span> 2(</em></strong> <input type="text" style="height: auto;" data-c="j + k - 1"> <span class="review"></span> <strong><em>)</em></strong></p><hr class="hidden-separator">Any integer with a factor of 2 is <strong><em>even</em></strong></div><div class="q-explanation"></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Complete the following to <strong><em>"prove that one more than the product of two consecutive odd integers is the square of an even number"</em></strong></p><p><span class="q-text-draggable draggable" draggable="true">2n²</span> <span class="q-text-draggable draggable" draggable="true">2n</span> <span class="q-text-draggable draggable" draggable="true">2n + 1</span> <span class="q-text-draggable draggable" draggable="true">4n²</span> <span class="q-text-draggable draggable" draggable="true">2n - 1</span></p></div><div class="q-answer"><p>Let <strong><em>2n - 1</em></strong> be the first odd integer</p><p><input type="text" style="height: auto;" data-c="2n + 1"> <span class="review"></span> is the next odd integer</p><p><strong><em>(2n - 1) (</em></strong> <input type="text" style="height: auto;" data-c="2n + 1"> <span class="review"></span> <strong><em>) + 1 <span class="math-tex">\(\equiv\)</span> </em></strong><input type="text" style="height: auto;" data-c="4n²"> <span class="review"></span> <strong><em>+ 2n - 2n - 1 + 1</em></strong></p><p><strong><em><span class="math-tex">\(\equiv\)</span> </em></strong><input type="text" style="height: auto;" data-c="4n²"> <span class="review"></span></p><p><span class="math-tex">\(\equiv\)</span> <em><strong>(</strong></em> <input type="text" style="height: auto;" data-c="2n"> <span class="review"></span> <strong><em>)²</em></strong></p><p><strong><em>2n</em></strong> is an even number</p></div><div class="q-explanation"><p>Finding the product means that we multiple the two numbers together</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="848"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Verify that <span class="math-tex">\(x^2\left(x+1\right)^2-\left(x-1\right)^2x^2=4x^3 \)</span> for x = 3</p> <p>b) Prove that <span class="math-tex">\(x^2\left(x+1\right)^2-\left(x-1\right)^2x^2\equiv4x^3 \)</span> for all x</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">b) Start with the left hand side. Expand brackets and simplify<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/deductive-proofs/esq_deductive-proofs1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/deductive-proofs/esq_deductive-proofs1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="849"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Verify that <span class="math-tex">\(\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}=\frac{2x+1}{x^2\left(x+1\right)^2} \)</span> for x = -2</p> <p>b) Prove that <span class="math-tex">\(\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}\equiv\frac{2x+1}{x^2\left(x+1\right)^2} \)</span> for all x</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content>b) Start with the left hand side, add the algebraic fractions together by making the denominators the same.</section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/deductive-proofs/esq_deductive-proofs2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/deductive-proofs/esq_deductive-proofs2.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="850"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Prove that the sum of three consecutive integers is divisible by 3</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Let the first integer be <strong><em>n</em></strong></p> <p><content>What would be the value of the next integer?</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/deductive-proofs/esq_deductive-proofs3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/deductive-proofs/esq_deductive-proofs3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="851"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Verify that x² - 4x + 5 is positive for x = -1</p> <p>b) Prove that x² - 4x + 5 is positive for all x</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">A square number cannot be negative. Is it posible to write this expression as a square?<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/deductive-proofs/esq_deductive-proofs4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/deductive-proofs/esq_deductive-proofs4.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="852"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Prove that the difference between the square of any two consecutive odd integers is divisible by 8</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>An odd number can be written as 2n - 1.</p> <p>A number is divisible by 8 if it has a factor of 8<content></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/deductive-proofs/esq_deductive-proofs5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/deductive-proofs/esq_deductive-proofs5.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 6</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="853"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a) Verify that <span class="math-tex">\(^3 C_1\ +\ ^3 C_2\ =\ ^4C_2\)</span></p> <p>b) Prove that <span class="math-tex">\(^{n-1} C_{r-1}\ +\ ^{n-1} C_r\ =\ ^nC_r\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="math-tex">\(^nC_r=\frac{n!}{(n-r)!r!}\)</span><content> </content></p> <p>Some careful manipulation is required for this proof. Consider that <span class="math-tex">\(r!=r\cdot(r-1)!\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/deductive-proofs/esq_deductive-proofs6.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/deductive-proofs/esq_deductive-proofs6.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="1733"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label 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