Question

If = {x: x is an integer such that 1B = {2, 4, 6, 8, 10, 12, 14, 16, 18}.
List down the elements of A’ and B’

Are sets A - B and B - A equal sets?

Find and compare the sets (AUB)’ and A’B’ and share your observations

Answer 1

GIVEN :

If $U={\{x: x is an integer such that 1<x<20}\}$ and $A={\{8, 9, 10, 11, 12, 13, 14}\}$ and $B={\{2, 4, 6, 8, 10, 12, 14, 16, 18}\}$

TO FIND :

(i) List down the elements of $A^\prime$ and $B^\prime$

(ii) Are sets A - B and B - A equal sets

(iii) Find and compare the sets $(AUB)^\prime$ and $A^\prime \cap B^\prime$ and share your observations

SOLUTION :

Given that the sets if $U={\{x: x is an integer such that 1<x<20}\}$ and $A={\{8, 9, 10, 11, 12, 13, 14}\}$ and $B={\{2, 4, 6, 8, 10, 12, 14, 16, 18}\}$

From the given the universal set can be written as

$U={\{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}\}$

(i) List down the elements of $A^\prime$ and $B^\prime$

$A^\prime=U-A$

$A^\prime={\{2,3,4,5,6,7,15,16,17,18,19}\}$

$B^\prime=U-B$

$B^\prime={\{3,5,7,9,11,13,15,17,19}\}$

(ii) Are sets A - B and B - A equal sets

Now $A-B={\{8,9,10,11,12,13,14}\} - {\{2,4,6,8,10,12,14,16,18}\}$

$A-B={\{9,11,13}\}$

$B-A={\{2,4,6,8,10,12,14,16,18}\}-{\{8,9,10,11,12,13,14}\}$

$B-A={\{2,4,6,16,18}\}$

From the above $A-B\neq B-A$

∴ A-B and B-A are not equal sets

(iii) Find and compare the sets $(AUB)^\prime$ and $A^\prime \cap B^\prime$ and share your observations

$A\cup B={\{8,9,10,11,12,13,14}\} U {\{2,4,6,8,10,12,14,16,18}\}$

$A\cup B={\{2,4,6,8,9,10,11,12,13,14,16,18}\}$

$(A U B)^\prime=U-AUB$

$(A U B)^\prime={\{3,5,7,15,17,19}\}$

We know $A^\prime={\{2,3,4,5,6,7,15,16,17,18,19}\}$ and $B^\prime={\{3,5,7,9,11,13,15,17,19}\}$

$A'\cap B'={\{3,5,7,15,17,19}\}$

∴ $(A U B)^\prime=A^\prime \cap B^\prime$

Both the sets $(A U B)^\prime$ and $A^\prime \cap B^\prime$ are equal sets.