Question

State, whether the following pairs of sets are equal or not:

(i) A=$\{2,4,6,8\}$ and $B=\{2n:n\in Nandn<5\}$

(ii) $M=\{x:x\in Wandx+3<8\}andN=\{y:y=2n-1,n\in Nandn<5\}$

(iii) $E=\{x:x^2+8x-9=0\}andF=\{1,-9\}$

(iv) $A=\{x:x\in ,x<3\}andB=\{y:y^2-3y+2=0\}$ Two sets are equal, if both the sets have same (identical) elements.

Answer 1

(i) A = $\{2,4,6,8\}$ B=A=$\{2n:n\in Nandn<5\}$

When n =1, 2n = $2\times 1=2$

When n=2, 2n =$2\times 2=4$

When n=3, 2n =$2\times 3=6$

When n=4, 2n =$2\times 4=8$

$\therefore$ B=$\{2,4,6,8\}$

$\text{Now we see that elements of sets A and B are the same (identical)}$

$\therefore$ Sets A and B are equal. Ans.

(ii) M = $\{x:x\in Wandx+3<8\}$

x+3<8 $\Rightarrow$

x<8-3 $\Rightarrow$

x<5 $\therefore$ M = $\{0,1,2,3,4\}$

N=$\{y:y=2n-1,n\in Nandn<5\}$

y=2n-1

When n=1, y=2 $\times$ 1-1 $\Rightarrow$ y = 2-1 = 1

When n=2, y=2 $\times$ 2-1 $\Rightarrow$ y =4-1=3

When n=3, y=2 $\times$ 3-1 $\Rightarrow$ y =6-1=5

When n=4, y=2 $\times$ 4-1 $\Rightarrow$ y =8-1=7

$\therefore$ N=$\{1,3,5,7\}$

$\text{Now we see that elements of sets M and N are not the same (identical)}.$ $\therefore$ Sets M and N are not equal. Ans.

(iii) $E=\{x:x^2+8x-9=0\}{x}^2+8x-9=0\Rightarrow x^2+9x-x-9=0\Rightarrow x(x+9)-1(x+9)=0\Rightarrow (x-1)(x+9)=0\therefore Eitherx+9=0Orx-1=0\Rightarrow x=-9\Rightarrow x=1\therefore E=\{-9,1\}F=\{1,-9\}\text{Now we see that the elements of sets E and F are the same (identical)}\therefore SetsEandFareequal.Ans.$

(iv) $A=\{x:x\in N,x<3\}=\{1,2\}B=\{y:y^2-3y+2=0\}{y}^2-3y+2=0\Rightarrow y^2-2y-y+2=0\Rightarrow y(y-2)-1(y-2)=0\Rightarrow (y-2)(y-1)=0\therefore Eithery-2=0ory-1=0\Rightarrow y=2\Rightarrow y=1\therefore B=\{1,2\}\text{Now we see that elements of sets A and B are the same (identical)}.\therefore SetsAandBareequal.Ans.$