Suppose we have a set, S, and that T is a subset of S, as shown in the diagram below.
If an element y is in T, then y must also be in S, because T, is a subset of S. Let's refer to this as Statement A:
A: If an element y is in T, then y is in S.
Statement B: The Contrapositive of A
Let the contrapositive of Statement A be Statement B,
B: If an element, y, is not in S, then it cannot be in T.
Is Statement B True?
Is Statement B true? Suppose we have a representative element y that is not in set S.
Since set T is inside set S, y cannot be in T. Statement B must be true.
B Follows from A
Since set T is a subset of S, any element that is not in S cannot be in T. Otherwise, T would not be a subset of S. So Statement B follows from A.
A Follows from B
Suppose now that we were only given Statement B:
B: If an element, y, is not in S, then it cannot be in T
From this statement we can derive Statement A:
- if an element, y, that is not in S cannot be in T, then T must be a subset of S.
- if T is a subset of S, then any element that is in T, must also be in S, which is Statement A.
Since Statement A follows from B, and B follows from A, these two statements are equivalent. That is, one statement implies the other.
In general, if one of the two statements are true, then the other must be true (a proof is beyond what is required for the ISM).