LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

OPERATION ON SETS - UNION OF SETS

**OPERATION
ON SETS –**

**(1) Union Of Sets –**

**The union of two sets A & B
is a set C formed by combining the elements of A & B. It contains all
elements in either of the sets A or B. Elements that are common to both A and B need to be listed only once in
set C. The symbol A ∪ B means
the union of A and B, and is read “A union B”.**

**
A ∪ B = {x ǀ x ∈ or x ∈ B or x ∈ both A & B}**

**Example.1) If A = {1, 2, 3}, and
B = {4, 5, 6}, A ∪ B = ?**

** Ans.) A ∪ B = {1,
2, 3, 4, 5, 6}**

**Example.2) If A = {p, q, r}, B =
{r, s, t, x, y, z}, then find A ∪ B**

**Ans.) A ∪ B = {p, q, r, s, t, x, y, z}**

**Example.3) If A = {factor of
12}, and B = {factors of 16} then find A ∪ B **

**Ans.) A = {factor of 12} = {1, 2, 3, 4, 6, 12}, **

** B = {factors of 16} = {1,
2, 4, 8, 16}, **

** and A ∪ B = {1,
2, 3, 4, 6, 8, 12, 16}**

**Example.4) If A = {2, 3, 5, 7,
11}, B = {1, 3, 5, 7, 9}, C = {0, 1, 2, 3} then prove (A ∪ B) ∪ C = A ∪ (B ∪ C)**

**Ans.) A ∪ B = {1, 2, 3, 5, 7, 9, 11}**

**(A ∪ B) ∪ C = {1, 2, 3, 5, 7, 9, 11} ∪ {0, 1, 2,
3}**

** = {0, 1, 2, 3, 5, 7, 9,
11}**

**On the other hand,**

**B ∪ C = {1, 3, 5, 7, 9} ∪ {0, 1, 2, 3} = {0, 1, 2, 3, 5, 7, 9}**

**A ∪ (B ∪ C) = {2, 3, 5, 7, 11} ∪
{0, 1, 2, 3, 5, 7, 9} **

** = {0, 1, 2, 3, 5, 7, 9, 11}**

**So, we can say that, (A ∪ B) ∪ C = A ∪ (B ∪ C) (Proved)**

**Example.5) Is it true that for any
sets A & B, P(A) ∪ P(B ) = P(A ∪ B) ? Justify your answer **

**Ans.) To prove, let us take an
example suppose, A = {p}, B = {q}.**

**Then, A ∪ B = {p, q}**

**So, P(A) = Set of subsets of A =
{ϕ, {p}}**

**P(B) = Set of Subsets of B = {ϕ,
{q}}**

**So, P(A) ∪ P(B) = {ϕ, {p},
{q}} ……………………..(i)**

**A ∪ B = {p, q}**

**=> P(A ∪ B) = set of subsets
of P(A ∪ B)**

**So, P(A ∪ B) = {ϕ, {p}, {q}, {p,
q}} ……………………..(ii)**

**From (i) & (ii)is clear
that P(A) ∪ P(B) ≠ P(A ∪ B) **

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