Proposition Let A and B be sets.
AΔ B = (A ∪ B) - (A∩B).
Let A and B be sets.
Suppose x∈AΔB. Thus x∈ (A-B)∪(B-A). This means either x∈A-B or x∈B-A. We consider both cases.
Suppose x∈A-B. So x∈A and x is not in B. Since x∈A we have x∈A∪B. Since x is not in B we have that x is not in A∩B. Thus x∈(A∪B) - (A∩B).
Suppose x∈ B-A. By a similar argument x∈ (A∪ B) - (A∩B).
Suppose x∈(A ∪B) - (A∩B). Then x∈ A ∪B and x is not in A ∩ B. This means that x is in A or in B but not in both. Thus either x is in A but not in B or x is in B but not in A. So x∈(A-B)∪(B-A). Therefore x∈A ΔB.
Therefore AΔB= (A∪B) - (A∩B).
QED