Question

prove A delta B = (A union B) - ( A union B)

Answer 1

Answer:

Prove that AΔB=(A∖B)∪(B∖A)

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The set AΔB consists of those elements that belong to exactly one of the sets A, B. Thus AΔB=(A∪B)∖(A∩B).

We show that AΔB=(A∖B)∪(B∖A), by showing that each set is contained in the other.

Let x∈AΔB. Then x∈A, x∉B or x∈B, x∉A. In the first case, x∈A∖B, and in the second case x∈B∖A. In any case, x∈(A∖B)∪(B∖A).

Now let x∈(A∖B)∪(B∖A). Then x∈A∖B or x∈B∖A. In the first case, x∈A, x∉B, and in the second case, x∈B, x∉A. Therefore x is in exactly one of the sets A, B. Hence x∈AΔB.

This proves AΔB=(A∪B)∖(A∩B)