Question

Which of the following pairs of sets are equal? Justify your answer.
$\left(i\right)X,$ the set of letters in $ALLOY$ and $B,$ the set of letters in $LOYAL$

$\left(ii\right)A=\{n:n\in Z$ and $n^2\le 4\}$ and $B=\{x:x\in R$ and $x^2-3x+2=0\}$

Answer 1

$\left(i\right)$ Step $1:$ Solve for elements of set $X.$
$X=\{A,L,O,Y\}$

Step $2:$ Solve for elements of set $B$
$B=\{L,O,Y,A\}$

Step $3:$ Comparing elements of set $X$ and set $B$
Every element of $X$ is same as every element of $B$

$\therefore X=B$

$\left(ii\right)$ Step $1:$ Solve for elements of set $A$
$A=\{n:n\in Z$ and $n^2\le 4\}$
$\therefore A=\{-2,-1,0,1,2\}$

Step $2:$ Solve for elements of set $B.$
$B=\{x:x\in R$ and $x^2-3x+2=0\}$
On solving $x^2-3x+2=0$
$x^2-2x-x+2=0$
$x(x-2)-1(x-2)=0$
$\Rightarrow (x-2)(x-1)=0$
$\Rightarrow x=1$ or $x=2$
$\therefore B=\{1,2\}$

Step 3: Comparing elements of set $X$ and set $B.$
Clearly $\{-2,-1,0\}$ is in set $A$ but not in set $B.$
$\therefore A\ne B$