Question

For all sets A and B, show that (A union B) - B =A - B​

Answer 1

Answer:

Step-by-step explanation:

“The stuff in A is the same as the stuff either: 1.) in A but not B or 2.) in both A and B.” Read this way, what you want to show is pretty clear, but perhaps you would like something more proof-y.

Consider the following (I’m using  ∨  for “or” and  ∧  for “and”):

(A−B)∪(A∩B)={x∣x∈A∧x∉B}∪{x∣x∈A∧x∈B}  (Definition of set difference, intersection)

={x∣(x∈A∧x∉B)∨(x∈A∧x∈B)}  (Definition of union)

={x∣x∈A∧(x∈B∨x∉B)}  (Distributivity of  ∧  over  ∨ )

={x∣x∈A}=A