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Question
If A is a subset of B , then prove that A union B is B .
If A is a subset of B , then prove that A union B is B .
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Solution
(i) Let A ⊆ B.
(ii) Let x ∈ A U B
(iii) Then x ∈ A or x ∈ B, by definition of union
(iv) By definition of a subset x ∈ A ⇒ x ∈ B
(v) Using (iii) and (iv), if x ∈ A, then x ∈ B. If x ∈ B, then clearly x ∈ B.
(vi) Thus given A ⊆ B, then A U B = B
(ii) Let x ∈ A U B
(iii) Then x ∈ A or x ∈ B, by definition of union
(iv) By definition of a subset x ∈ A ⇒ x ∈ B
(v) Using (iii) and (iv), if x ∈ A, then x ∈ B. If x ∈ B, then clearly x ∈ B.
(vi) Thus given A ⊆ B, then A U B = B
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