Mastering Mathematics Smartly
by Wee Wen Shih

A unique self-help website that provides comprehensive coverage of mathematics at A-level & beyond, written in a student-friendly style.

Section 1

Please refer to these sites (1 | 2) to study the key concepts.

Section 2

We shall look at some worked examples that require the application of key results you came across in Section 1.

Worked example 1: Show that A \cap B \subseteq A.
Let x \in A \cap B, we show that x \in A.
Now x \in A \cap B implies x \in A and x \in B.
So x \in A. Hence A \cap B \subseteq A.

Note: It is also true that A \cap B \subseteq B.

Worked example 2: Show that A \cap (A - B)^{\text{C}} \subseteq B.
First, we note that A - B = A \cap B^{\text{C}}.
By the complement of an intersection, we obtain (A - B)^{\text{C}} = (A \cap B^{\text{C}})^{\text{C}} = A^{\text{C}} \cup B.
Now LHS is simplified to A \cap(A^{\text{C}} \cup B).
By the distributive law of intersection over union, we obtain A \cap(A^{\text{C}} \cup B) = (A \cap A^{\text{C}}) \cup (A \cap B) = A \cap B.
By the Note in example 1, we arrive at the desired result.