Discrete Math: prove an intersection from a given

[JackRyan]

Discrete Math: prove B intersection A = A, given A-B = null set

1. Problem Statement:
Prove B $\cap$ A = A, given A-B = ∅ (empty set)

The Attempt at a Solution

xε(B$\cap$A) => xεB and xεA => Logic given A-B = ∅ => xεA

I tried using A-B = A$\cap$!B for xε(A$\cap$!B)=∅ => xεA and x not in !B or x not in A and Xε!B

I am unsure how to fill in that logic section and prove that B$\cap$A=A

 

[Dr. T]

The first direction is fine. If x is in A$\bigcap$B then, certainly, x is in A (no need to use the given statement). Now suppose that x is in A and proceed by contradiction (to show that x is in A$\bigcap$B). If x is not also in B determine what that implies about the given statement.