Question

Prove that : A – (A – B)= A ∩ B

Answer 1

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$\bold{\longmapsto}$Step-by-step explanation:

A-B means everything in A except for anything in A∩B

A-B=A∩B' (A intersect B complement)

pick an element x

let x∈(A-B)

therefore x∈A but x∉B

x∉B means x∈B'

x∈A and x∈B'

x∈(A∩B')

x∈(A-B)

therefore A-B=A∩B'

Here we have LHS=RHS.[⭐Hence proved ⭐]

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Answer 2

Answer:

A \ (A \ B) is the part of A that does not intersect with (A \ B). ~~A ∩ B. That would bring us right back to the original portion that intersected with B. Therefore A ∩ B = A ∩ B.

Step-by-step explanation:

To prove that A\(A\B) = A∩B, we have to show that (1) every element x of the set

A\(A\B) is an element of the set A∩B, i.e, A\(A\B) is a subset of A∩B, and (2) every element y of the set A∩B is an element of the set A\(A\B).

(a) Suppose x ∈ A\(A\B) this implies that x ∈ A & x ∉ A\B i.e, either:

x ∈ A & (x ∈ A & x ∈ B) which implies that x ∈ A∩B, …. or

x ∈ A & (x ∉ A & x ∉ B) which leads to a contradiction as you can’t write x ∈ A and x ∉ A in the same statement.Therefore A\(A\B) is a subset of A∩B

(b) Suppose y ∈ A∩B this implies that y ∈ A & y ∈ B, thus y ∉ A\B. Here you can argue and say that if y ∈ A and y ∉ A\B then y ∈ A\(A\B).Therefore A∩B is a subset of A\(A\B)

Conclusion: we showed that the two sets A\(A\B) and A∩B are subset of each other, therefore: A\(A\B) = A∩B