Chapter 6.2, Problem 20ES

To determine

To prove:

Find the mistake in the following “proof” that for all sets $A,B\text{ and }C$, if $A\subseteq B$ and $B\subseteq C$ then $A\subseteq C$.

Suppose $A,B\text{ and }C$ are sets such that $A\subseteq B$ and $B\subseteq C$. Since $A\subseteq B$, there is an element $x$ such that $x\in A\text{ and }x\in B$. Since $B\subseteq C$, there is an element $x$ such that $x\in A\text{ and }x\in C$. Hence there is an element $x$ such that $x\in A\text{ and }x\in C$ and so $A\subseteq C$.