Kenneth Rosen Edition 7 Exercise 2.1 Question 17 (Page No. 126)

Question

Suppose that $A, B,$ and $C$ are sets such that $A \subseteq B$ and $B \subseteq C.$ show that $A \subseteq C.$

Comments

We can show it using examples and that seems pretty obvious but it is not the correct way to prove something.

I would like to show it using contradiction.

 

Assume that  $A$  $\subseteq$  $C$    is false. So there must be an extra element in  $A$ , say $x$  which does not belongs to the  $C$ .  Now it is given that  $B$  $\subseteq$  $C$ , and   $A$   $\subseteq$   $B$  that means that extra element   $x$   must be present in  $B$ . Eventually element must belongs to    $C$  which contradicts our assumption.  So our assumption must be wrong which states that   $A$   $\subseteq$   $C$  is true.

 

Note : As per me , it is mentioned in the begining of the    $Rosen$   that the answers to both  $Odd$   and  $Even$   quetions are available , you may refer that  pdf.