Use the element method to prove that a set is empty, prove or disprove that a set is a subset of another set, or prove that two sets are equal.

Question

Use the element method to prove that if Q ⊆ R, R ⊆ S, and

S ⊆ Q, then Q = R.

Answer 1

HI here I believe by the "element" method is they want you start by thinking about an arbitrary element in Q.

1. So you start with Let x be an element in Q, and then using the definition of subsets you can show that x must also be an element of R and S.

Since you are trying to prove equality you must prove the subsets in both directions ie.

1) Q is a subset of R

AND

2) R is a subset of Q.

I talked about part 1 and you can do something similar for part 2.