Question

Q2. If​ ε ={x:xisanintegersuchthat1(i) List down the elements of A’ and B’
(ii) Are sets A - B and B - A equal sets?
(iii) Find and compare the sets (AUB)’ and A’ ⋂ B’ and share your observations

Answer 1

GIVEN :

If $\varepsilon={\{x: x is an integer such that 1<x<20}\}$ 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 $(A\cup B)^\prime$ and $A^\prime \cap B^\prime$ and share your observations

SOLUTION :

Given that the sets $\varepsilon={\{x: x is an integer such that 1<x<20}\}$ 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

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

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

$A^\prime=\varepsilon-A$

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

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

$B^\prime=\varepsilon-B$

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

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

(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 we have that

$A-B\neq B-A$

∴ A-B and B-A are not equal sets.

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

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

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

Now $(A\cup B)^\prime=\varepsilon-(A\cup B)$

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

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

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

$={\{3,5,7,15,17,19}\}$

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

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

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

Answer 2

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