Let A, B and C are sets. Prove that ≤ is transitive. (5 marks)

To show that $\subset$ is transitive we are to prove the following statement:

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

Let x be an element of A.

Since $x\in A$ and $A\subset B$ then $x\in B$.

Since $x\in B$ and $B\subset C$ then $x\in C$.

Therefore $\forall x\ x\in A\to x\in C$ and this means $A\subset C$ .

The statement is proved.