Cartesian Product

Q.

Let $S$ be the set of all integer solutions, $(x,y,z)$, of the system of equations

$$\begin{gathered} x-2y+5z=0 \\ -2x+4y+z=0 \\ -7x+14y+9z=0 \end{gathered}$$

such that $15\le x^2+y^2+z^2\le 150$.

Then, the number of elements in the set $S$ is equal to:

Q. Let $X=\{x:x\in N\text{and}-3<x<4\}$ and $Y=\{x:x\in N\text{and}{x}^2+x-6\le 0\}.$ Then the number of elements in $(X\times Y)\cap (Y\times X)$ is

1. $0$
2. $1$
3. $2$
4. $4$

Q. If $A\times B=\left\{\right(1,1),(1,2),(1,3),(3,1),(3,2),(3,3),(4,1),(4,2),(4,3\left)\right\}$ then

1. $A=\{1,2,3\}$
2. $B=\{2,3,4\}$
3. $A=\{1,3,4\}$
4. $B=\{1,2,3\}$

Q. If $A=\{y:x^2+y^2<1,x\in N\}$ and $B=\{y:x^2+y^2=4,x\in N\},$ then $n(A\times B)$ is

1. $0$
2. $1$
3. $3$
4. infinite

Q. If $A\times B=\left\{\right(1,1),(1,2),(1,3),(3,1),(3,2),(3,3),(4,1),(4,2),(4,3\left)\right\}$ then

1. $A=\{1,2,3\}$
2. $B=\{2,3,4\}$
3. $A=\{1,3,4\}$
4. $B=\{1,2,3\}$

Q.

If $A=\{1,2,3\}$ and $B=\{4,5\},$ then $A\times B=$ ___.

1. $\left\{\right(1,2),(1,3),(2,3),(4,5\left)\right\}$

2. $\left\{\right(4,1),(4,2),(4,3),(5,1),(5,2),(5,3\left)\right\}$

3. $\left\{\right(1,4),(1,5),(2,4),(2,5),(3,4),(3,5\left)\right\}$

4. $\left\{\right(1,4),(1,5),(2,4),(2,5\left)\right\}$

Q. If $A=\{x:x\text{is an even number and}0<x<10\}$ and $B=\{2,3,5,7\}$, then the number of elements in $A\cup B$ is

Q. If $A$ and $B$ are two sets such that $A\times B=\left\{\right(1,1),(1,2),(1,3),(2,1),(2,2),(2,3\left)\right\}$, then $B$ is

1. $\{1,2,3\}$
2. $\{1,2\}$
3. $\{2,3\}$
4. none of these

Q. Let $Y=\{1,2,3,4,5\},A=\{1,2\},B=\{3,4,5\}.$ If $A\times B$ denotes the cartesian product of sets $A$ and $B,$ then $(Y\times A)\cap (Y\times B)$ is

1. $Y$
2. $A$
3. $B$
4. Null set

Q. Let $A$ be a non-empty set such that $A\times A$ has $16$ elements among which three elements are found to be $(a,b),(b,c)$ and $(c,d)$ and $I_A$ be the identity relation on $A,$ then

1. $A=\{a,b,c\}$
2. $I_A=\left\{\right(a,a),(b,b),(c,c\left)\right\}$
3. $A=\{a,b,c,d\}$
4. $I_A=\left\{\right(a,a),(b,b),(c,c),(d,d\left)\right\}$

Q. If $A\times B=\left\{\right(a,1),(a,2),(a,3),(b,1),(b,2),(b,3\left)\right\},$ then $n\left(A\right)+n\left(B\right)=$ .

1. 4
2. 5
3. 6
4. 7

Q. If $n\left(A\right)$ denotes the number of elements in set $A$ and if $n\left(A\right)=4,n\left(B\right)=5$ and $n(A\cap B)=3$ then $n\left[\right(A\times B)\cap (B\times A\left)\right]$ is

Q. If $A=\{x:x-2=y,y<2,x,y\in W\}$ and $B=\{y:x+3=y,x<3,x,y\in W\},$ then

1. $A=\{2,3\}$
2. $B=\{3,4,5\}$
3. $(A-B)\cap (B-A)=\varphi$
4. $n\left(\right(A\Delta B)\times (B\Delta A\left)\right)=9$