If $A\subseteq B$ and $B\subseteq C$ then $A\subseteq C$.

Question

can someone please correct me if i made some mistakes here on ma solution. i used Lynx hea Problem D

1.) Give the definition of $S\subseteq T$ for general sets $S$ and $T$.

Solution

The set $S$ is a subset of $T$ if and only if every element of S is also an element of T.

2.) Prove that for all sets $A, B, C$ if $A\subseteq B$ and $B\subseteq C$ then $A\subseteq C$.

Solution

Suppose that $x\in A$. Because $A\subseteq B$, this implies that $x\in B$ Because $B\subseteq C$ we see that $x\in C$. Because $x \in A$ it implies that $x\in C$. It follows that $A\subseteq C$.

Answer 1

Your solutions are both good, but I have a nitpick with the first one:

You shouldn't define terms using "if and only if."

Ideally you would just use "if" here. The most commonly given reason for this is that making a definition is really a different thing from a biconditional statement. You'll find this is the case for most of the literature.