Kinds Of Function ( Reflexive, Symmetric , Transitive Antisymmetric ) By Ishrat Hayat Malik

<div dir="ltr" style="text-align: left;" trbidi="on"> <h2 style="text-align: center;"> <span style="background-color: cyan; font-weight: normal;">Kinds Of Function ( Reflexive, Symmetric , Transitive Antisymmetric ) By Ishrat Hayat Malik</span></h2> <div> <span style="background-color: cyan;"></span><br /> <h3 style="text-align: left;"> <span style="background-color: cyan;"> <b><u>Kinds Of Function In Discrete Structure<div class="separator" style="clear: both; text-align: center;"> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEji0wFKx__ntFlL_U8BJ_hoim-TlIDAuqhLpSXnYwB2QmeD5WYMC7a5bk-X3PwJ4kdqrGFpyMa2liaxCp8gq1ePxClXGMXBZytIWD_cyTuHlDVweP5Lu9pr4U0bXtzKCLPvBeEqubef5cFy/s1600/slide_1.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEji0wFKx__ntFlL_U8BJ_hoim-TlIDAuqhLpSXnYwB2QmeD5WYMC7a5bk-X3PwJ4kdqrGFpyMa2liaxCp8gq1ePxClXGMXBZytIWD_cyTuHlDVweP5Lu9pr4U0bXtzKCLPvBeEqubef5cFy/s320/slide_1.jpg" width="320" /></a></div> </u></b></span></h3> <span style="background-color: cyan;"> </span> <h3 style="text-align: left;"> <span style="background-color: cyan;"> <span style="font-weight: normal;">1) Reflexive<br />2) Symmetric<br />3) Transitive<br />4) Anti-Symmetric</span></span></h3> <span style="background-color: cyan;"> </span><span style="background-color: cyan;"><h3 style="text-align: left;"> <b><u>• Reflexive:</u></b> The relation R on {1,2,3} given by</h3> <h3 style="text-align: left;"> <span style="font-weight: normal;">R = {(1,1), (2,2), (2,3), (3,3)} is reflexive. (All<br />loops are present.)</span></h3> <h3 style="text-align: left;"> <b><u>• Symmetric</u></b>: The relation R on {1,2,3} given by</h3> <h3 style="text-align: left;"> <span style="font-weight: normal;">R = {(1,1), (1,2), (2,1), (1,3), (3,1)} is symmetric.<br />(All paths are 2-way.)</span></h3> <h3 style="text-align: left;"> •<b><u> Transitive:</u></b> The relation R on {1,2,3} given by</h3> <h3> <span style="font-weight: normal;">R = {(1,1), (1,2), (2,1), (2,2), (2,3), (1,3)} is<br />transitive. (If I can get from one point to another<br />in 2 steps, then I can get there in 1 step.)</span></h3> <h3 style="text-align: left;"> <b><u>Anti-Symmetric</u></b>  </h3> <h3 style="text-align: left;"> <span style="font-weight: normal;">R = {(1,1), (1,2), (3,2), (3,3)} is antisymmetric. • Is every relation symmetric or anti-symmetric? • No! Consider R = {(1,2), (2,1), (1,3)}</span></h3> <div> <h3 style="text-align: center;"> <u style="font-weight: normal;">Violations of the Properties</u></h3> <h3 style="text-align: left;"> <span style="font-weight: normal;">• Why is R = {(1,1), (2,2), (3,3)} not reflexive on<br />{1,2,3,4}?<br />Because (4,4) is missing.<br />• Why is R = {(1,2), (2,1), (3,1)} not symmetric?<br />Because (1,3) is missing.<br />• Why is R = {(1,2), (2,3), (1,3), (2,1)} not<br />transitive?<br />Because (1,1) and (2,2) are missing.<br />• Is {(1,1), (2,2), (3,3)} symmetric? transitive?<br /> Yes! Yes!</span></h3> </div> <div> <br /></div> </span><span style="background-color: cyan;"><div style="font-weight: normal;"> <br /></div> </span></div> </div>

Kinds Of Function ( Reflexive, Symmetric , Transitive Antisymmetric ) By Ishrat Hayat Malik

Kinds Of Function ( Reflexive, Symmetric , Transitive Antisymmetric ) By Ishrat Hayat Malik Kinds Of Function In Discrete Structure ...

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Induction In Discrete Structure By Ishrat Hayat Malik

<div dir="ltr" style="text-align: left;" trbidi="on"> <div dir="ltr" style="text-align: left;" trbidi="on"> <h2 style="text-align: center;"> <span style="background-color: cyan; font-weight: normal;">Induction In Discrete Structure By (Ishrat Hayat Malik)</span></h2> <div> <span style="background-color: white;"><b>Define</b></span><span style="background-color: white; font-weight: normal;">: </span> Induction is a mathematical method which is used to show a statement, a system or a theorem is real for each natural number.<br /> The method entails two steps to prove a statement, as mentioned beneath −<br /> <br /></div> <div> </div> <div style="text-align: center;"> <b><u>Example</u></b></div> <div style="text-align: center;"> <div style="font-weight: bold; text-decoration-line: underline;"> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPngheJlCmenVSLn_1ZVoql3t9sDoYg_wzOBXMG8vuSC6m_y0rERyiACDLBcp3zNi0Pz9WH0tDzxNk4jXIKLdPfUIvt_7qgtv8cFfnfbpweoSgK29WzGttkTN4GbcHmmOUjfMw98hcalnw/s1600/slide_1.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPngheJlCmenVSLn_1ZVoql3t9sDoYg_wzOBXMG8vuSC6m_y0rERyiACDLBcp3zNi0Pz9WH0tDzxNk4jXIKLdPfUIvt_7qgtv8cFfnfbpweoSgK29WzGttkTN4GbcHmmOUjfMw98hcalnw/s200/slide_1.jpg" width="200" /></a>Problem 1</div> <div> 3n−13n−1 is a multiple of 2 for n = 1, 2, ...</div> <div> <br /></div> <div> <b>Solution</b></div> <div> <br /></div> <div> <b>Step 1</b> − For n=1,31−1=3−1=2n=1,31−1=3−1=2 which is a multiple of 2</div> <div> <br /></div> <div> <b>Step 2 −</b> Let us assume 3n−13n−1 is true for n=kn=k, Hence, 3k−13k−1 is true (It is an assumption)</div> <div> <br /></div> <div> We have to prove that 3k+1−13k+1−1 is also a multiple of 2</div> <div> <br /></div> <div> 3k+1−1=3×3k−1=(2×3k)+(3k−1)3k+1−1=3×3k−1=(2×3k)+(3k−1)</div> <div> The first part (2×3k)(2×3k) is certain to be a multiple of 2 and the second part (3k−1)(3k−1) is also true as our previous assumption.</div> <div> <br /></div> <div> Hence, 3k+1–13k+1–1 is a multiple of 2.</div> <div> <br /></div> <div> So, it is proved that 3n–13n–1 is a multiple of 2.</div> <div> <br /></div> <div> <b><u>Problem 2</u></b></div> <div> 1+3+5+...+(2n−1)=n21+3+5+...+(2n−1)=n2 for n=1,2,…n=1,2,…</div> <div> <b>Solution</b></div> <div> <br /></div> <div> <b>Step 1</b> − For n=1,1=12n=1,1=12, Hence, step 1 is satisfied.</div> <div> <br /></div> <div> <b>Step 2 </b>− Let us assume the statement is true for n=kn=k.</div> <div> <br /></div> <div> Hence, 1+3+5+⋯+(2k−1)=k21+3+5+⋯+(2k−1)=k2 is true (It is an assumption)</div> <div> <br /></div> <div> We have to prove that 1+3+5+...+(2(k+1)−1)=(k+1)21+3+5+...+(2(k+1)−1)=(k+1)2 also holds</div> <div> <br /></div> <div> 1+3+5+⋯+(2(k+1)−1)1+3+5+⋯+(2(k+1)−1)</div> <div> =1+3+5+⋯+(2k+2−1)=1+3+5+⋯+(2k+2−1)</div> <div> =1+3+5+⋯+(2k+1)=1+3+5+⋯+(2k+1)</div> <div> =1+3+5+⋯+(2k−1)+(2k+1)=1+3+5+⋯+(2k−1)+(2k+1)</div> <div> =k2+(2k+1)=k2+(2k+1)</div> <div> =(k+1)2=(k+1)2</div> <div> So, 1+3+5+⋯+(2(k+1)−1)=(k+1)21+3+5+⋯+(2(k+1)−1)=(k+1)2 hold which satisfies the step 2.</div> <div> <br /></div> <div> Hence, 1+3+5+⋯+(2n−1)=n21+3+5+⋯+(2n−1)=n2 is proved.</div> <div> <br /></div> <div> <b><u>Problem 3</u></b></div> <div> Prove that (ab)n=anbn(ab)n=anbn is true for every natural number nn</div> <div> Solution</div> <div> <br /></div> <div> <b>Step 1 </b>− For n=1,(ab)1=a1b1=abn=1,(ab)1=a1b1=ab, Hence, step 1 is satisfied.</div> <div> <br /></div> <div> <b>Step 2 − </b>Let us assume the statement is true for n=kn=k, Hence, (ab)k=akbk(ab)k=akbk is true (It is an assumption).</div> <div> <br /></div> <div> We have to prove that (ab)k+1=ak+1bk+1(ab)k+1=ak+1bk+1 also hold</div> <div> <br /></div> <div> Given, (ab)k=akbk(ab)k=akbk</div> <div> <br /></div> <div> Or, (ab)k(ab)=(akbk)(ab)(ab)k(ab)=(akbk)(ab) [Multiplying both side by 'ab']</div> <div> <br /></div> <div> Or, (ab)k+1=(aak)(bbk)(ab)k+1=(aak)(bbk)</div> <div> <br /></div> <div> Or, (ab)k+1=(ak+1bk+1)(ab)k+1=(ak+1bk+1)</div> <div> <br /></div> <div> Hence, step 2 is proved.</div> <div> <br /></div> <div> So, (ab)n=anbn(ab)n=anbn is true for every natural number n<br /> <br /> <b><span style="color: red;"><u>Video Tutorial In Urdu</u></span></b><br /> <br /></div> </div> </div> <iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/bieb8A4be5A" width="560"></iframe> </div>

Induction In Discrete Structure By Ishrat Hayat Malik

Induction In Discrete Structure By (Ishrat Hayat Malik) Define :   Induction is a mathematical method which is used to show a statement...

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Array Definition In C (Discrete Structure) Ishrat Hayat Malik

<div dir="ltr" style="text-align: left;" trbidi="on"> <h2 style="text-align: center;"> <span style="background-color: cyan; font-weight: normal;">Array Definition In C (Discrete Structure) Ishrat Hayat Malik</span></h2> <div> <h3 style="text-align: left;"> <span style="background-color: white; font-weight: normal;"><u>Array:</u></span></h3> <div class="separator" style="clear: both; text-align: center;"> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvOF08L2GPb_OUNdBtDss7IpRaPcVWVuRiTd1jKRF8iJozcRAouQ_Gr3uOVaLvkIbu3JLtMvgVcJeg7ANCXX9Q_VVukwgR3HnClHX1eeP0hP5s94GumrkLmII1mppck5KUrcbme3NF5ji_/s1600/C-Array.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvOF08L2GPb_OUNdBtDss7IpRaPcVWVuRiTd1jKRF8iJozcRAouQ_Gr3uOVaLvkIbu3JLtMvgVcJeg7ANCXX9Q_VVukwgR3HnClHX1eeP0hP5s94GumrkLmII1mppck5KUrcbme3NF5ji_/s200/C-Array.png" width="200" /></a></div> <div> <span style="background-color: white; font-weight: normal;"><u><br /></u></span></div> <div style="text-align: left;"> </div> <ol style="text-align: left;"> <li>Array Is used to have the large Number of Collection .</li> <li>It is represented by Sqaure.bracket [ ] which contain the size of element. </li> <li>it contain always homogenenous data </li> <li>.Addresses are always on memory Location.</li> <li>Start Address Of Array with 0.</li> </ol> </div> <div style="text-align: left;"> <span style="background-color: white;"><br /></span></div> </div>

Array Definition In C (Discrete Structure) Ishrat Hayat Malik

Array Definition In C (Discrete Structure) Ishrat Hayat Malik Array: Array Is used to have the large Number of Collection . It...

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Forms Of Set (Discrete Structure)

<div dir="ltr" style="text-align: left;" trbidi="on"> <h2 style="text-align: center;"> <span style="background-color: cyan;">Forms Of Sets</span></h2> <div class="separator" style="clear: both; text-align: center;"> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbZ8OU9MSCjsc2T_dIV0IKDcj5sqJmLsL2tZKbuHhuDuKH0pUUBCtp6ewmNcxkSqo4zttMpQB0GTP3LKCH6oMRd8LEXI5f7qbCujEWSPyG4TXuuyDQgRG2TGDfFdhY5r987u-euF1_VRpI/s1600/maths-sets-ppt-5-638.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbZ8OU9MSCjsc2T_dIV0IKDcj5sqJmLsL2tZKbuHhuDuKH0pUUBCtp6ewmNcxkSqo4zttMpQB0GTP3LKCH6oMRd8LEXI5f7qbCujEWSPyG4TXuuyDQgRG2TGDfFdhY5r987u-euF1_VRpI/s320/maths-sets-ppt-5-638.jpg" width="320" /></a></div> <div> <span style="background-color: cyan;"></span><br /> <div> <span style="background-color: cyan;"><b>1) Tabular/Roster Form</b></span></div> <span style="background-color: cyan;"> </span> <br /> <div> <span style="background-color: cyan;"><b>2) Descriptive Form</b></span></div> <span style="background-color: cyan;"> </span> <div> <span style="background-color: cyan;"><b>3) Set Builder Form</b></span></div> <span style="background-color: cyan;"> <div> <b><u>Roster form or tabular form:</u></b></div> <div> In this, elements of the set are listed within the pair of brackets { } and are separated by commas. For example:</div> <div> (i) Let N denote the set of first five natural numbers.</div> <div> Therefore, N = {1, 2, 3, 4, 5}        → Roster Form</div> <div> <b><u>Descriptive Form:</u></b></div> <div> State in words the elements of sets.</div> <div> Example:</div> <div> A=A= Set of first five natural numbers.</div> <div> B=B= Set of positive even integers less or equal to fifty.</div> <div> C=C= Set of positive odd integers.</div> <div> <b><u>Set Builder Form:</u></b></div> <div> In set Builder form all the element of the set possesses with the single common property to become the member of the set</div> <div> A={x:x∈N} N = natural numbers</div> <div> <br /></div> </span></div> </div>

Forms Of Set (Discrete Structure)

Forms Of Sets 1) Tabular/Roster Form 2) Descriptive Form 3) Set Builder Form Roster form or tabular form: In this, ele...

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