Question

Show that if \[A \subset B\], \[C - B \subset C - A\].

Answer 1

Hint:
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'.