Textbook Problem

In 3 and 4, supply explanations of the steps in the given proofs.

Theorem: For all sets A, B, and C, if $A\subseteq B$ , $B\subseteq C$ then $A\subseteq C$ .
Proof:

Statement	Explanation
Suppose A, B, and C are any sets such that

$A\subseteq B$
and

$B\subseteq C$
.
starting point
We must show that

$A\subseteq C$
.
conclusion to be shown
Let x be any element in A.
start of an element proof
Then x is in B.
(a) .
It follows that x is in C. (b) . Thus every element in A is in C.
since x could be any element of A Therefore,

$A\subseteq C$
[as was to be shown].
(c) .