Q. If the set $A$ has $3$ elements and the set $B=\{3,4,5\}$ then find the number of elements in $(A\times B)$?

Q. If $A\times B=\left\{\right(a,x),(a,y),(b,x),(b,y\left)\right\}$ Find $A$ and $B$.

Q. Let $A=\{1,2\},B=\{1,2,3,4\},C=\{5,6\}$ and $D=\{5,6,7,8\}$ Verify that
(i) $A\times (B\cap C)=(A\times B)\cap (A\times C)$
(ii) $A\times C$ is a subset of $B\times D$

Q. If $G=\{7,8\}$ and $H=\{5,4,2\}$, find $G\times H$ and $H\times G$.

Q. Let $A=\{1,2\}$ and $B=\{3,4\}$. Write $A\times B$ and find how many subsets will $A\times B$ have? List them.

Q. Let $A$ and $B$ be two sets such that $n\left(A\right)=3$ and $n\left(B\right)=2$. If $(x,1),(y,2),(z,1)$ are in $A\times B$ find $A$ and $B$ where $x,y$ and $z$ are distinct elements

Q. The cartesian product $A\times A$ has $9$ elements among which are found $(-1,0)$ and $(0,1)$. Find the set $A$ and the remaining elements of $A\times A$

Q. If $=A\{-1,1\}$ then find $A\times A\times A$.

Q. State whether each of the following statements are true or false. If the statement is false rewrite the given statement correctly
(i) If $P=\{m,n\}$ and $Q=\{n,m\}$ then $P\times Q=\left\{\right(m,n),(n,m\left)\right\}$
(ii) If $A$ and $B$ are non-empty sets then $A\times B$ is a non-empty set of ordered pairs $(x,y)$ such that $x\in A$ and $y\in B$
(iii) If $A=\{1,2\},B=\{3,4\}$ then $A\times (B\cap \varphi )=\varphi$

Q. If $(\frac{x}{3}+1,y-\frac{2}{3})=(\frac{5}{3},\frac{1}{3})$ find the values of x and y

Q. Let $A=\{1,2,3,4,6\}$ and $R$ be the relation on $A$ defined by $\left\{\right(a,b):a,b\in A,b$ is exactly divisible by $a\}$
(i) Write $R$ in roster form
(ii) Find the domain of $R$
(iii) Find the range of $R$

Q. Define a relation $R$ on the set $N$ of natural numbers by $R=\left\{\right(x,y):y=x+5,x$ is a natural number less than $4;x,y\in N\}$. Depict this relationship using roster form. Write down the domain and the range.

Q. The figure shows a relationship between the sets $P$ and $Q$. Write this relation in
(i) in set-builder form (ii) roster form

Q. Let $A=\{1,2,3,....,14\}$. Define a relation $R$ from $A$ to $A$ by $R=\left\{\right(x,y):3x-y=0$ where $x,y\in A\}$. Write down its domain, co-domain and range.

Q. Let $R$ be the relation on $Z$ defined by $R=$ $\left\{\right(a,b):a,b\in Z,a-b$ is an integer$\}$. Find the domain and range of $R.$

Q. Determine the domain and range of the relation $R$ defined by $R=\left\{\right(x,x+5):x\in \{0,1,2,3,4,5\}\}$

Q.

Write the relation $R=\left\{\right(x,x^3):x$ is a prime number less than $10\}$in roster form.

Q. Let $A=\{x,y,z\}$ and $B=\{1,2\}$. Find the number of relations from $A$ to $B$.

Q. $A=\{1,2,3,5\}$ and $B=\{4,6,9\}$. Define a relation $R$ from $A$ to $B$ by $R=\left\{\right(x,y):$ the difference between $x$ and $y$ is odd $x\in A,y\in B\}$. Write $R$ in roster form

Q. Which of the following relations are functions? Give reasons.