Question

Let $P=\left\{1,-1\right\}$ $,$ $Q=\left\{x:x^2=4,x\in \mathrm{\mathbb{Z}}\right\}$ and $R=\left\{-2,-1,1\right\}$ . State which of the following is true and which is false$.$

$P\leftrightarrow Q$

• A. True
• B. False

Answer 1

The correct option is A True

It is given that $P=\left\{1,-1\right\}$$,$

$Q=\left\{x:x^2=4,x\in \mathrm{\mathbb{Z}}\right\}$$,$

$R=\left\{-2,-1,1\right\}$

$P=\left\{1,-1\right\}$

Number of elements in set $P$ is $n\left(P\right)=2$

The Roster form of the set $Q=\left\{x:x^2=4,x\in \mathrm{\mathbb{Z}}\right\}$ is given below

$x^2=4,x\in \mathrm{\mathbb{Z}}$

$x^2-4=0$

$\left(x-2\right)\left(x+2\right)=0$

$x=2$or $x=-2$

$Q=\left\{2,-2\right\}$

Number of elements in set $Q$ is $n\left(Q\right)=2$

$R=\left\{-2,-1,1\right\}$

Number of elements in the set $R$ is given below

$n\left(R\right)=2$

For any two equivalent sets number of elements in the sets must be equal.

$n\left(P\right)=2$ and $n\left(Q\right)=2$ therefore the number of elements in sets $P$ and $Q$ are equal.

Hence$,$ the given statement $P\leftrightarrow Q$ is true.