If A and B are sets with A⊆B the a) A ∪B = B, b) A∩B = A.
a) Let A⊆B
Let x ∈ A∪B
Then x ∈ A or x ∈ B .....(i)
By definition of union we have
x ∈ A ⇒ x ∈ B .....(ii)
Using (i) and (ii), if x ∈ A, then x ∈ b. If x ∈ B, then clearly x ∈ B.
Thus given A⊆ B, then A∪B = B.
b) First assume A∩B which implies that x ∈ A and x ∈ B.
Hence we can implies the
A∩ B ⊆ A (we started with the element in A∩B and concluded that it is in A) ------(i)
Now, let x ∈ A which implies x ∈ B ( as A⊆ B)
Hence, x ∈ A∩B, which in turn imply
A ⊆ A∩B -----(ii)
From (i) and (ii) we get
A∩B = A
Therefore if A and B are sets with A⊆B, then A ∩B = B and A∩B = A.
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