Question

Show that if $A \subset B$, $C - B \subset C - A$.

Answer 1

Let the element be x belonging to set A as $A \subset B$ and so x will also belong to B. And so now for the element to be in C it should not be in set B and similarly in set A. Hence, from there we can state the following to show as given above.
For the given $A \subset B$ , it means that all the elements of A are present in set B.

Complete step by step solution:
As given that $A \subset B$
Let $x \in C - B$…(1)
$\Rightarrow x \in C,x \notin B$
As $A \subset B$,
Now if an element belongs to set A then it will also be there in set B as A is subset of B, and if an element is not present in set B then it will also be not present in set A, so we can say that
$\Rightarrow x \in C,x \notin A$
So, we have, $x \in C - A$…(2)
Hence from 1 and 2, we can state that,
$\therefore C - B \subset C - A$

Hence, if $A \subset B$ then $C - B \subset C - A$ is true.

Note:
A set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion.
Also let the belonging element in both the sets be selected with proper logic. And understand the reason behind each and every step.
Therefore, $X \in A$ will be read as 'x belongs to set A'.