Intersection of Sets

The INTERSECTION of two sets A and B is the set of all elements that are in BOTH A and B, denoted $A \cap B$. Formally, $A\cap B = \{x/x\in A \text{ and } x\in B\}$. In a Venn diagram, this is just the overlapping (lens-shaped) region shared by the two circles (Fig. 5.5). For example, if $A=\{1,3,5,7,9\}$ and $B=\{1,2,3,4,5,6,7,8\}$, then $A\cap B = \{1,3,5,7\}$. As another example, if $A=\{x/x\in N, x \text{ is a factor of } 12\}=\{1,2,3,4,6,12\}$ and $B=\{x/x\in N, x \text{ is a factor of } 18\}=\{1,2,3,6,9,18\}$, then $A\cap B$ = the common factors of 12 and 18 = $\{1,2,3,6\}$. If instead $A=\{1,3,5,7,9\}$ and $B=\{2,4,6,8,10\}$, then $A\cap B=\phi$ — when two sets share no elements at all, $A\cap B=\phi$, and A and B are called DISJOINT sets.

Properties of intersection: (i) $A\cap B = B\cap A$ (Commutativity); (ii) $(A\cap B)\cap C = A\cap(B\cap C)$ (Associativity); (iii) $\phi\cap A = \phi$; (iv) $A\cap A = A$ (Idempotent law); (v) $A\cap A' = \phi$ (a set and its complement share nothing); (vi) if $A\subset B$ then $A\cap B = A$; (vii) $U\cap A = A$ (Identity for intersection); (viii) $(A\cap B)\subset A$ and $(A\cap B)\subset B$; (ix) Distributivity: $A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$, and $A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$. …