prove that if A is a proper subset of B and B is a subset of C, then A is a proper subset of C.?

Answer 1

Take an element a, from the set A. Since A is a proper subset of B, a is in B. Since B is a subset of C, a is in C. Thus A is a subset of C.

However, there is some b in B but not in A. We know b is in C, since B is a subset of C. Since b is in C but not in A, A is a proper subset of C.

Answer 2

Since A is a proper subset of B, by definition A is not all of B. Since A is a proper subset of B, B has at least 1 element since if B were empty, then A would be empty, but the empty set is not a proper subset of itself.

B is a set with 1 or more elements and B is a subset (proper or not) of C. If A is empty then A is a proper subset of C, since the empty set is a proper subset of every set except itself.

Suppose A has one or more elements. All of those elements are in B. B is a subset of C, so C contains all of B (and may in fact be B). Hence, those elements of B that are in C which are also form A means that A is a subset of C. Since by hypothesis, A is a proper subset of B, A cannot be all of C, and so is a proper subset of C.

HTH

Charles

Answer 3

First, we show that A is a subset of C (we'll do the "proper" part later).

If A is empty, then it's a subset of every set (including C), by definition.

If A is not empty, then consider an arbitrary element θ in A. Since A is a subset of B, it must be true that θ is also in B.

Since B is a subset of C, every element of B (including θ) is also in C.

Therefore, every element θ of A, is also an element of C. By definition, that makes A a subset of C.

Now for the "proper" part.

If A is a proper subset of B, then by definition there is at least one element φ, which is in B but is not in A.

If B is a subset of C, then by definition φ is also in C.

That means φ is in C but not in A. Combined with the previously established fact that A is a subset of B, this means that A is a proper subset of C.

Answer 4

A is a proper subset of B: so every element of A is an element of B and there is an element of B (say x) which is not in A.
Since B is a subset of C, every element of B is an element of C. Since every element of A is an element of B, it must also be an element of C. So A is a subset of C. Also, specifically x is an element of B, hence also an element of C. Since x is not in A, A must be a proper subset of C.