Which of the following are pairs of equal sets?

Question:

Which of the following are pairs of equal sets?

$E=\left\{x: x \in Z, x^{2} \leq 4\right\}$ and $F=\left\{x: x \in Z, x^{2}=4\right\}$

 

Solution:

Equal Sets = Two sets A and B are said to be equal if they have exactly the

same elements & we write A = B

We have

$E=\left\{x: x \in Z, x^{2} \leq 4\right\}$

Here, $x \in Z$ and $x^{2} \leq 4$

If $x=-2$, then $x^{2}=(-2)^{2}=4=4$

If $x=-1$, then $x^{2}=(-1)^{2}=1<4$

If $x=0$, then $x^{2}=(0)^{2}=0<4$

If $x=1$, then $x^{2}=(1)^{2}=1<4$

If $x=2$, then $x^{2}=(2)^{2}=4=4$

So,

$E=\{-2,-1,0,1,2\}$

and $F=\left\{x: x \in Z, x^{2}=4\right\}$

Here, $x \in Z$ and $x^{2}=4$

If $x=-2$, then $x^{2}=(-2)^{2}=4=4$

If $x=2$, then $x^{2}=(2)^{2}=4=4$

So, $F=\{-2,2\}$

∴ E ≠ F because the elements in the both the sets are not equal.

 

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