Difference and Symmetric Difference of Sets

The DIFFERENCE of set A and set B is the set of elements that are in A but NOT in B, denoted $A-B$ (also written $A\cap B'$). Formally, $A-B = \{x/x\in A, x\notin B\}$ (shaded in Fig. 5.6). Similarly, $B-A = \{y/y\in B, y\notin A\}$. For example, if $A=\{x/x\in N, x \text{ is a factor of } 6\}=\{1,2,3,6\}$ and $B=\{x/x\in N, x \text{ is a factor of } 8\}=\{1,2,4,8\}$, then $A-B=\{3,6\}$ and $B-A=\{4,8\}$.

Notes on difference: (i) $A-B\subseteq A$ and $B-A\subseteq B$; (ii) the three sets $A-B$, $A\cap B$, and $B-A$ are mutually disjoint — no two of them share any element; (iii) $A-B=A\cap B'$ and $B-A=A'\cap B$; (iv) $A\cup B = (A-B)\cup(A\cap B)\cup(B-A)$ — the union splits cleanly into these three disjoint pieces (Fig. 5.7).

SYMMETRIC DIFFERENCE: $(A-B)\cup(B-A)$ is called the symmetric difference of A and B, denoted $A \Delta B$ — it collects everything that is in exactly one of A, B, but not in both. For example, if $A=\{4,5,6,7,8\}$ and $B=\{3,5,6,8,9\}$, then $A-B=\{4,7\}$, $B-A=\{3,9\}$, so $A\Delta B = \{4,7\}\cup\{3,9\} = \{3,4,7,9\}$. …