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## Statement **A**

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 elementyis inT, thenyis inS.

## Statement **B**: The Contrapositive of **A**

Let the **contrapositive** of Statement **A** be Statement **B**,

B: If an element,y, is not inS, then it cannot be inT.

## 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 inS, then it cannot be inT

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**.

## Equivalence

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).

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