<h2 id="success-cartesian-product--relationssuccess"><Success> Cartesian product & relations</Success></h2> <div style='float: left; width:50%;'> </div> <div style='float: right; width:50%;'> </div>

<h3 id="successexamplesuccess"><Success>Example</Success></h3> <div style='float: left; width:90%;font-size:30px;''> <p>Let A={1,2,3}, B={3,4} & C={4,5,6}. Find</p> <p>(i) A$\times$(B∩C) <br> (ii) (A$\times$B)∩(A$\times$C)<br> (iii) A$\times$(B∪C)<br> (iv) (A$\times$B)∪(A$\times$C)</p> </div> <div style='float: right; width:10%;'> </div>

<h3 id="successsolutionsuccess"><Success>Solution</Success></h3> <div style='float: left; width:90%;font-size:30px;''> <p>(i) A={1,2,3}, B∩C={4}</p> <p>A$\times$(B∩C)={(1,4),(2,4),(3,4)}</p> <p>(ii) A$\times$B={(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)}</p> <p>A$\times$C={(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)}.</p> <p>(A$\times$B)∩(A$\times$C) = {(1,4),(2,4),(3,4)}</p> <p>A$\times$(B∩C) = (A$\times$B)∩(A$\times$C)</p> </div> <div style='float: right; width:10%;'> </div>

<h3 id="successsolutionsuccess-1"><Success>Solution</Success></h3> <div style='float: left; width:90%;font-size:30px;'> <p>(iii) B={3,4}, C={4,5,6}</p> <p>∴ B∪C = {3,4,5,6}</p> <p>A$\times$(B∪C) = {(1,3),(1,4),(1,5),(1,6),(2,3)(2,4),(2,5),(2,6),(3,3)(3,4),(3,5),(3,6)}.</p> <p>A$\times$B & A$\times$C were calculated earlier.</p> <p>(A$\times$B)∪(A$\times$C) ={(1,3),(1,4),(2,3)(2,4),(3,3)(3,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6)}</p> <p>A$\times$(B∪C) = (A$\times$B)∪(A$\times$C).</p> </div> <div style='float: right; width:10%;'> </div>

<h3 id="successexamplesuccess-1"><Success>Example</Success></h3> <div style='float: left; width:90%;font-size:30px;'> <p>Let P={a,b,c}. Find the set P$\times$P$\times$P.</p> </div> <div style='float: right; width:10%;'> </div>

<h3 id="successsolutionsuccess-2"><Success>Solution</Success></h3> <div style='float: left; width:90%;font-size:30px;'> <ul> <li> <p>P$\times$P$\times$P = {(a,a,a),(a,a,b),(a,a,c),(a,b,a),(a,b,b),(a,b,c),(a,c,a),(a,c,b),(a,c,c),(b,a,a),(b,a,b),(b,a,c),(b,b,a),(b,b,b),(b,b,c),(b,c,a),(b,c,b),(b,c,c),(c,a,a),(c,a,b),(c,a,c),(c,b,a),(c,b,b),(c,b,c),(c,c,a),(c,c,b),(c,c,c)}.</p> </li> <li> <p>(P$\times$P$\times$P) = 27.</p> </li> </ul> </div> <div style='float: right; width:10%;'> </div>

#### <Success>Example on cartesian product of two set</Success> <div style='float: left; width:90%;font-size:30px;'> <p>1.) {($x,y$) | $x$ and $y$ are real numbers & $x^2+y^2=1$}⊆$\R^2$</p> <p>This is not a cartesian product of two sets.</p> <p>2.) A={Ramu,Babu,Ramesh,Kumar,Siva}.</p> <p>B={Laxmi,Manju,Mani}.</p> <p>R={(Ramesh,Mani),(Ramu,Manju),(Babu,Laxmi)}.</p> <p>Ramesh & Mani - are married<br> Ramu & Manju - are married<br> Babu & Laxmi - are married</p> </div> <div style='float: right; width:10%;'> </div>

### <Success>Relation</Success> <div style='float: left; width:90%;font-size:30px;'> <p>${\bold{\text{Definition:}}}$</p> <p>Let A and B be any two nonempty sets. A relation R from A to B is a nonempty subset of A$\times$B.</p> </div> <div style='float: right; width:10%;'> </div>

### <Success>Remarks</Success> <div style='float: left; width:90%;font-size:30px;'> <p>#(A $\times$ B) = #(A).#(B)</p> <ul> <li> <p>How many relations are possible between A & B?<br> How many subset of A$\times$B are possible?</p> </li> <li> <p>If A is any nonempty set, then the number of possible subset of A is 2^#(A).</p> </li> </ul> <p>∴ The no. of possible nonempty subsets of A$\times$B is<br> 2^{#(A$\times$B)}$-$1</p> <ul> <li>∴ The total no. of possible relations between of A$\times$B is 2^{#(A$\times$B)} -1</li> </ul> </div> <div style='float: right; width:10%;'> </div>

### <Success>Universal relation</Success> <div style='float: left; width:90%;font-size:30px;'> <p>Let A={1,2,3}</p> <p>$\quad$ B={2,3,4}</p> <ul> <li> <p>Notice that A$\times$B is itself a relation between A and B. This relation is called the universal relation.</p> </li> <li> <p>For any two sets A & B, A$\times$B is called the universal relation.</p> </li> </ul> </div> <div style='float: right; width:10%;'> </div>

### <Success>Example</Success> <div style='float: left; width:60%;font-size:30px;'> <p>A={1,2,3}</p> <p>B={2,3,4}</p> <p>R={(1,2),(1,3),(2,2),(2,3),(3,4)}.</p> </div> <div style='float: right; width:40%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/baf64247-7b67-4e95-b986-74e9eeb2ea00/public"></p> </div>

### <Success>Example</Success> <div style='float: left; width:60%;font-size:30px;'> <p>A={Ramu, Babu, Ramesh, Kumar, Siva}</p> <p>B={Laxmi, Manju, Mani}</p> <p>R={(Ramesh,Mani),(Ramu,Manju),(Babu,Laxmi)}</p> </div> <div style='float: right; width:40%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/cdb67f7a-ae90-4655-13a7-f49b42ba4400/public"></p> </div>

### <Success>Example</Success> <div style='float: left; width:50%;font-size:30px;'> <p>A={1,2,3,4,5}</p> <p>B={-1,0,4,9,25}</p> <p><strong>R={(x,y) ∈ A × B | y=x²}.</strong></p> <p>R={(2,4),(3,9),(5,25)}</p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/45ebc0b3-9e7a-4016-040d-2459c769f500/public"></p> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/00c4b9ed-bd2f-4bea-b074-bb43830f6600/public"></p> </div>

### <Success>Definitions</Success> <div style='float: left; width:90%;font-size:30px;'> <ul> <li> <p>Let A and B be two nonempty sets and let R be a relation between A and B.</p> </li> <li> <p>The <strong>domain(R)</strong> is the set of all first elements from the ordered pairs in R.</p> </li> <li> <p>The set B is called as the <strong>codomain of R</strong>.</p> </li> <li> <p>The set of all second elements from the ordered pairs in R is called the <strong>range of R</strong>.</p> </li> </ul> </div> <div style='float: right; width:10%;'> </div>

### <Success>Example</Success> <div style='float: left; width:50%;font-size:30px;'> <p>Let A = {1,2,3,4,5,6}</p> <p><strong>R = {(x,y)∈A × A | y=x+1}.</strong></p> <p>R={(1,2),(2,3),(3,4),(4,5),(5,6)}</p> <p><strong>Domain(R)</strong> = {1,2,3,4,5}</p> <p><strong>Codomain(R)</strong> = {1,2,3,4,5,6}</p> <p><strong>Range(R)</strong> = {2,3,4,5,6}.</p> </div> <div style='float: right; width:50%;'> <!----> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/82986b58-9c46-4916-12de-31de0789df00/public"></p> </div>

### <Success> Example</Success> <div style='float: left; width:50%;font-size:30px;'> <p>A={4,9,10,25}</p> <p>B={-5,-3,-2,1,2,3,5}</p> <p><strong>R={(x,y)∈A × B | x is the square of y}.</strong></p> <p>R={(4,-2),(4,2),(9,-3),(9,3),(25,-5),(25,5)}.</p> <p><strong>Domain(R)</strong> = {4,9,25}</p> <p><strong>Codomain(R)</strong> = {-5,-3,-2,1,2,3,5}</p> <p><strong>Range(R)</strong> = {-3,-2,-5,2,3,5}.</p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/5ec4ffbd-ddb3-4952-30c6-b32b264cdc00/public"></p> </div> <div style='float: right; width:49%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/008c044c-adc8-4691-b1a0-1a41a93bd500/public"></p> </div>

### <Success>Example</Success> <div style='float: left; width:90%;font-size:30px;'> <p>#(A)=4 $\quad \quad$ #(B)=7</p> <p>#(A × B)=4 × 7=28.</p> <p>∴ The no. of possible relations from A to B is 2²⁸-1.</p> </div> <div style='float: right; width:10%;'> </div>

### <Success> Example</Success> <div style='float: left; width:90%;text-align:left;font-size:30px;'> <p>Consider the set of all natural numbers.Define R={(n,m)∈ $\N$ × $\N$ | m=n+5}.</p> <ul> <li> <p>$\N $ - set of all natural numbers.</p> </li> <li> <p>R is an infinite set.</p> </li> <li> <p>R = {(n,n+5) | n is a natural number}.</p> </li> <li> <p>R’= {(n,m)∈ $\N$ × $\N$ | m=n+5 & n≤4}.</p> </li> </ul> <p>= {(1,6),(2,7),(3,8),(4,9)}.</p> </div> <div style='float: right; width:10%;'> </div>

### <Success> Example</Success> <div style='float: left; width:60%;font-size:30px;'> <p>Let</p> <p>A ={1,2,3,5}</p> <p>B ={4,6,9}</p> <p>R={(x, y) $\in$ A × B $\mid$ the difference between x & y is an odd number }.</p> <p>= {(1,4),(1,6),(2,9),(3,4),(3,6),(5,4),(5,6)}.</p> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/913e9d59-a162-4a97-3fb4-2106d24d9400/public"></p> </div>

### <Success>Example</Success> <div style='float: left; width:90%;font-size:30px;'> <p>A={1,2,3,5}</p> <p><strong>R={(x,y)∈ A × A | the difference between x and y is an odd number}.</strong></p> <p>={(1,2),(2,1),(2,3),(2,5),(3,2),(5,2)}</p> <p>Whenever (x,y)∈R $\rightarrow$ (y,x)∈R.</p> </div> <div style='float: right; width:10%;'> </div>

### <Success>Symmetric relation</Success> <div style='float: left; width:90%;font-size:30px;'> <p><strong>Definition:</strong> Let A be a nonempty set and let R be a relation on A ,i.e., R is a nonempty subset of AxA. We say that R is a Symmetric.</p> <p>if (x,y)∈R $\implies$ (y,x)∈R.</p> </div> <div style='float: right; width:10%;'> </div>

### <Success>Thank you</Success> <div style='float: left; width:90%;font-size:30px;'> </div> <div style='float: right; width:10%;'> </div>