### <Success>Relation \& function: lecture 4</Success> <div style='float: left; width:100%;'> </div>

### <Success>Equivalence classes</Success> <div style='float: left; width:100%;font-size:30px;'> <ul> <li>Let $X$ be any set and let $R$ be a relation from $X$ to $X$. Let $R$ be an equivalence relation.</li> <li>Let $x \in X$. Then <strong>equivalence class of $x$</strong> will be,<br> $[x]_R = \{y∈X : (x,y) ∈R\}$.</li> </ul> </div> ====== <h3 id="successtheoremsuccess"><Success>Theorem</Success></h3> <div style='float: left; width:100%;font-size:30px;'> <p>Let $x,y ∈X$ with $x\neq y$.<br> Suppose that $[x]_R \neq [y] _R$.<br> <strong>Claim</strong>: $[x]_R ∩ [y]_R = \phi$.</p> </div>

### <Success> Proof</Success> <div style='float: left;text-align:left; width:100%;font-size:30px;'> <p>Suppose not, i.e,there exists $z ∈ [x]_R ∩ [y]_R $.</p> <ul> <li>$z∈[x]_R \ \& \ z∈[y]_R$</li> <li>$\Rightarrow (z, x)∈R \ \& \ (z, y)∈R$</li> <li>$\Rightarrow (x, z)∈R \ \& \ (z, y)∈R \ $ (∵ $R$ is an equivalence relation)</li> <li>$\Rightarrow (x,y)∈R$ (∵ $R$ is transitive)</li> <li>$\Rightarrow y∈[x]_R $</li> <li>$∴ [y]_R ⊆ [x]_R$</li> <li>By symmetry of $x \ \& \ y$ we have $[y]_R =[x]_R$.</li> <li>Thus $[x]_R = [y]_R$</li> <li>Or $[x]_R ∩ [y]_R =\phi$</li> </ul> </div>

### <Success>Example</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>$\Z$ = Set of all integers.<br> $R=\{(m,n)∈ \Z \times \Z | m≡n (\text{mod} \ 9)\}$. Write down all the equivalence classes for the given relation.</p> </div>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;'> <p>$1∈ \Z$<br> $[1]_R = \{m|m-1 ≡ 0 \ (\text{mod} \ 9)\}$<br> $m-1≡0 (\text{mod} \ 9)$<br> Equivalently, $m-1$ is divisible by $9$.</p> <p>$10$ is one such number.<br> $\{1,10,19,28,….\} = \{9n+1|n∈ \Z \}$</p> <p>$[1]_R = \{1,10,19,28,…\}$</p> <p>$[0]_R = \{0,9,18,27,…\} $</p> <p>Note that there are exactly 9 equivalence classes.<br> $\{[0]_R, [1]_R, [2]_R, [3]_R, [4]_R, [5]_R, [6]_R, [7]_R, [8]_R \}$.</p> </div>

### <Success>Example</Success> <div style='float: left; width:100%;font-size:30px;'> <p>$R=\biggl\{((x_1, y_1),(x_2, y_2))∈ \R^2$\ $\{(0,0)\}\times \R^2$\ $\{(0,0)\} \mid \text{there exists a non zero scalar}| α ∈ \R \ \text{such that} \ (x_1, y_1) = α (x_2, y_2)\biggl \}$.<br> Write down the equivalence class for the given relation.</p> </div>

### <Success>Solution</Success> <div style='float: left; width:100%;font-size:30px;'> <p>Suppose $(x_1, y_1)∈$ $\R^2$ \ $\{(0,0)\}$.</p> <p>$((x_2, y_2), (x_1, y_1))∈ R$</p> <p>$\Rightarrow (x_2, y_2) = α (x_1, y_1) \ \text{for some} \ α∈ \R \ \text{with} \ α≠0$.</p> <p>$\Rightarrow (x_2, y_2) \ \& \ (x_1, y_1) \ \text{lie on the same line passing through the origin}.$</p> <p>$\text{Thus} \ [(x_1, y_1)]_R \ \text{is the straight line passing through the origin}$<br> $\text{with the origin removed}.$</p> </div>

### <Success>Functions</Success> <div style='float: left; width:100%;font-size:30px;'> <p><strong>Definition</strong> A function from a nonempty set $A$ to a nonempty set $B$ is a relation from $A$ to $B$ with the following properties:<br> i) Domain $(f) = A$.</p> <p>ii) Whenever $(a,b_1)∈f \ \text{and} \ (a, b_2)∈f$<br> $\Rightarrow b_1 = b_2$</p> </div>

### <Success>Example</Success> <div style='float: left;text-align:left; width:50%;font-size:30px;'> <p>$A= \{-5,-3,-2,0,1,3,5\}$<br> $B= \{ 1,2,4,9,16,24,30\}$</p> <p>$R =\{(b,a)∈ B\times A \mid b=a^2 \}$.<br> Check if $R$ is a function or not.</p> <p><strong>Solution:</strong></p> <ul> <li>Some elements of $B$ are not related to $A$. e.g. $2,16,24$ and $30$</li> <li>$(9,-3)∈R \ \& \ (9, 3)∈R$</li> <li>$∴ R$ is not a function.</li> </ul> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/7360cd5b-45ec-42f8-3144-77683e638c00/public"></p> </div> <div style='float: right; width:49%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/124c52eb-cae4-4224-4f8a-5beb6a27e900/public"></p> </div>

### <Success>Example</Success> <div style='float: left; width:50%;font-size:30px;'> <p>$A=\{1,2,3,4,5,6\} \ \text{and} \ R = \{(a,b)∈ AxA | b= a+1 \}$ <br> Check if $R$ is a function or not.</p> <p><strong>Solution:</strong></p> <ul> <li>Domain $(R) = \{1,2,3,4,5\}≠A$</li> <li>$∴ R$ is not a function.</li> </ul> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/20d489d7-d399-4911-1812-75010bc7f800/public"></p> </div> <div style='float: right; width:49%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/5d289c11-9dae-48e7-8630-b9f402111300/public"></p> </div>

### <Success>Example</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>$f:\R → \R$</p> <p>$f = \{(x,y) | y=x^2\}$. Check if $f$ is a function or not.</p> <p><strong>Solution:</strong></p> <ul> <li>Domain$(f) =\R$.</li> <li>Suppose $(x, y_1)= (x, y_2)∈f$</li> <li>$(x, y_1) ∈f \Rightarrow y_1 = x^2$</li> <li>$(x, y_2) ∈f \Rightarrow y_2 = x^2$</li> <li>$\Rightarrow y_1 = y_2$.</li> <li>$∴$ f is a function.</li> </ul> </div>

### <Success>Notation</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <ul> <li>$f:A \rightarrow B$ is function <br> i.e, the relation $‘f’$ is a function from a non empty set $A$ to the nonempty set $B.$</li> <li>If the pair $(a,b) ∈ f$ this will be denoted as $b=f(a).$</li> </ul> <p>======</p> <h3 id="successrange-of-a-functionsuccess"><Success>range of a function</Success></h3> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <ul> <li>Suppose $f:A \rightarrow B$ is a function.</li> <li>Range $(f) = {b∈B: b= f(a)}.$</li> </ul> </div>

### <Success>Example</Success> <div style='float: left;text-align:left; width:100%;font-size:30px;'> <p>$f:\R \rightarrow \R$<br> $f(x)=x^2$<br> Find the range of the function.</p> <p><strong>Solution:</strong></p> <ul> <li>Let $y∈ \R$ with $y \ge 0.$</li> <li>Let us assume that $y≠0 \Rightarrow y>0$.</li> <li>$\Rightarrow$ there are exactly two square roots of $y$, namely $+\sqrt{y} \ \ \& \ -\sqrt{y}.$</li> <li>Let $x=+\sqrt{y} \ \Rightarrow \ y=x^2$</li> <li>Thus, Range$(f) = \{y∈\R | y \ge 0 \}.$</li> </ul> </div>

### <Success>Example</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>$f:\Z \rightarrow \Z$<br> $f(n) =n+1$ <br> Find the range of the function.</p> <p><strong>Solution:</strong></p> <ul> <li>Range $(f) = \Z$.</li> <li>Let $m ∈ \Z$</li> <li>Let $n=m-1 \Rightarrow f(n) =n+1 =(m+1) +1 =m.$</li> <li>Thus Range $(f) =\Z.$</li> </ul> </div>

### <Success>Identity function</Success> <div style='float: left; 50%;font-size:30px;'> <ul> <li>$f:\R \rightarrow \R$<br> $f(x) =x$</li> <li>Domain $(f) = \R$</li> <li>Range $(f) = \R$</li> </ul> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/53943bdf-eb0a-4ff0-bdc9-52fe9ba53200/public"></p> </div> <div style='float: right; width:49%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/711ae6b9-2a26-4d92-be1d-093e4abb9800/public"></p> </div>

### <Success>$f(x)=x^2 \ \text{and} \ f(x)= \sin x$</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <ol> <li>$f(x) =x^2$</li> </ol> <ul> <li>Domain $(f) = \R$</li> <li>Range $(f) = \{y \mid y \ge 0\}$</li> </ul> <ol start="2"> <li>$f(x) = \sin x$</li> </ol> <ul> <li>Domain $(f) = \R$</li> <li>Range $(f) = [-1,1]$</li> </ul> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/01cde3ea-1d9e-4dde-4cf6-1f2bb1084300/public"></p> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/2e281701-f2d1-4e86-ae99-bb252d5ae900/public"></p> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/2da0f678-1509-4300-cdc7-f1d43dcda400/public"></p> </div>

### <Success>$f(x)=|x| \ \text{and} \ f(x)=x^3$</Success> <div style='float: left; text-align:left; width:50%;font-size:30px;'> <ol start="3"> <li>$f(x) =|x|$</li> </ol> <ul> <li>Domain $(f) =\R$</li> <li>Range $(f) = \{ y∈ \R | y≥0\}.$</li> </ul> <ol start="4"> <li>$f(x) =x^3$</li> </ol> <ul> <li>Domain $(f) =\R$</li> <li>Range $(f) = \R$.</li> </ul> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/8cde806c-8f77-4d9a-d3e1-ec5057ead800/public"></p> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/3e13a80c-5e89-42bc-5868-62fb6cf24800/public"></p> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/e46be078-b280-4dc1-6dae-2f39ebefab00/public"></p> </div>

###### <Success>$\text{Relation between} \ f(A∩B) \\\ \text{and} \ f(A)∩f(B)$</Success> <div style='float: left;text-align:left; width:100%;font-size:30px;'> <p>$f:X \rightarrow Y$<br> Suppose $A,B ⊆ X$.<br> Find the relation between $f(A∩B)$ and $f(A)∩f(B)$.</p> <p><strong>Solution:</strong></p> <p>$f:\R → \R \ \ \text{and} \ f(x) =x^2.$</p> <ul> <li>$A=[0,4] ; B=[-4, 0]$</li> <li>$A∩B = {0}.$</li> <li>$f(A) = [0,16] ;f(B) = [0,16] ;f(A∩B) ={0}.$</li> <li>$∴ f(A∩B) ⊆ f(A)∩f(B).$</li> </ul> </div>

###### <Success>$\text{Relation between} \ f(A∩B) \\\ \text{and} \ f(A)∩f(B)$</Success> <div style='float:left;text-align:left; width:100%;font-size:30px;'> <ul> <li>Claim: For any $A,B ⊆ X,$<br> $f(A∩B) ⊆ f(A)∩f(B)$</li> <li>Let $y ∈ f(A∩B)$<br> $\Rightarrow$ there exists $x∈ A∩B$ such that $f(x)=y.$</li> <li>$x∈ A∩B \Rightarrow x∈A \ \& \ x∈B$</li> <li>$\Rightarrow f(x) ∈ f(A) \ \& \ f(x) ∈ f(B)$</li> <li>$\Rightarrow f(x) ∈ f(A) ∩ f(B).$</li> <li>$\Rightarrow y ∈ f(A) ∩ f(B).$</li> <li>Thus $f(A∩B) ⊆ f(A) ∩ f(B).$</li> </ul> </div>

#### <Success>$\text{Relation between} \ f(AUB) \ \text{and} \\\ f(A)Uf(B)$</Success> <div style='float: left; text-align:left; width:100%;font-size:30px;'> <p>$f:X \rightarrow Y$ Suppose $A,B ⊆ x.$<br> Find the relation between $f(AUB)$ and $(A) U f(B)$.</p> <p><strong>Solution:</strong></p> <ul> <li>Claim: $f(AUB) = f(A) U f(B).$</li> <li>Let $y ∈ f(AUB)$<br> $\Rightarrow$ there exists $x∈AUB$ such that $y=f(x).$</li> <li>$x∈AUB \ \Rightarrow \ x∈A \ \text{or} \ x∈B$</li> <li>$\Rightarrow f(x)∈f(A) \ \text{or} \ f(x)∈f(B)$</li> <li>$\Rightarrow f(x) ∈ f(A) U f(B)$ $\Rightarrow y ∈ f(A) U f(B)$</li> <li>$∴ f(AUB) ⊆ f(A) U f(B).$</li> </ul> </div>

#### <Success>$\text{Relation between} \ f(AUB) \ \text{and} \\\ f(A)Uf(B)$</Success> <div style='float: left;text-align:left; width:100%;font-size:30px;'> <ul> <li>Let $y ∈ f(A) U f(B)$</li> <li>$\Rightarrow y ∈ f(A) \ \text{or} \ y ∈ f(B)$</li> <li>$y ∈ f(A) \Rightarrow \ \text{there exists} \ x∈A \ \text{such that} \ y=f(x)$</li> <li>$y ∈ f(B) \Rightarrow \ \text{there exists} \ x’∈B \ \text{such that} y=f(x’).$</li> <li>$x∈A \Rightarrow x∈ AUB \Rightarrow f(x) ∈ f (AUB)$</li> <li>$x’∈B \Rightarrow x’∈ AUB \Rightarrow f(x’) ∈ f (AUB)$</li> <li>$∴ y = f(x) = f(x’) ∈ f (AUB)$</li> <li>$∴ f(A) U f(B) ⊆ f (AUB)$</li> <li>$∴ f(A) U f(B) = f (AUB)$</li> </ul> </div>

<div> <h2><Success>Thank you</Success></h2> </div>