# Problems on Union of Sets

Solved problems on union of sets are given below to get a fair idea how to find the union of two or more sets.

We know, the union of two or more sets is a set which contains all the elements in those sets.

Solved problems on union of sets:

1. Let A = {x : x is a natural number and a factor of 18} and B = {x : x is a natural number and less than 6}. Find A ∪ B.

Solution:

A = {1, 2, 3, 6, 9, 18}

B = {1, 2, 3, 4, 5}

Therefore, A ∪ B = {1, 2, 3, 4, 5, 6, 9, 18}

2. Let A = {0, 1, 2, 3, 4, 5}, B = {2, 4, 6, 8} and C = {1, 3, 5, 7}

Verify (A ∪ B) ∪ C = A ∪ (B ∪ C)

Solution:

(A ∪ B) ∪ C = A ∪ (B ∪ C)

L.H.S. = (A ∪ B) ∪ C

A ∪ B = {0, 1, 2, 3, 4, 5, 6, 8}

(A ∪ B) ∪ C = {0, 1, 2, 3, 4, 5, 6, 7, 8}     ……………….. (1)

R.H.S. = A ∪ (B ∪ C)

B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

A ∪ (B ∪ C) = {0, 1, 2, 3, 4, 5, 6, 7, 8}     ……………….. (2)

Therefore, from (1) and (2), we conclude that;

(A ∪ B) ∪ C = A ∪ (B ∪ C)  [verified]

More worked-out problems on union of sets to find the union of three sets.

3. Let X = {1, 2, 3, 4}, Y = {2, 3, 5} and Z = {4, 5, 6}.

(i) Verify X ∪ Y = Y ∪ X

(ii) Verify (X ∪ Y) ∪ Z = X ∪ (Y ∪ Z)

Solution:

(i) X ∪ Y = Y ∪ X

L.H.S = X ∪ Y

= {1, 2, 3, 4} ∪ {2, 3, 4} = {1, 2, 3, 4, 5}

R.H.S. = Y ∪ X

= {2, 3, 5} U {1, 2, 3, 4} = {2, 3, 5, 1, 4}

Therefore, X ∪ Y = Y ∪ X  [verified]

(ii) (X ∪ Y) ∪ Z = X ∪ (Y ∪ Z)

L.H.S. = (X ∪ Y) ∪ Z

X ∪ Y = {1, 2, 3, 4} U {2, 3, 5}

= {1, 2, 3, 4, 5}

Now (X ∪ Y) ∪ Z

= {1, 2, 3, 4, 5, 6} {4, 5, 6}

= {1, 2, 3, 4, 5, 6}

R.H.S. = X U (Y ∪ Z)

Y ∪ Z = {2, 3, 5} ∪ {4, 5, 6}

= {2, 3, 4, 5, 6}

X ∪ (Y ∪ Z) = {1, 2, 3, 4} ∪ {2, 3, 4, 5, 6}

Therefore, (X ∪ Y) ∪ Z = X ∪ (Y ∪ Z)  [verified]

Set Theory

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