Difference of Sets

Fig 1.8 is the standard Venn diagram for the difference of two sets. The rectangle $U$ represents the universal set. Inside it are two overlapping circles, labelled $A$ (on the left) and $B$ (on the right). The region that is shaded is the part of $A$ that does not overlap with $B$ — that is, the crescent-shaped portion of $A$ lying entirely outside $B$. An arrow points to this shaded region, and it is labelled $A - B$.

The physical idea is simple: you take everything in $A$, and then you remove whatever also belongs to $B$. What remains is the difference. The diagram makes it visually clear that $A - B$ and $B - A$ are completely different regions — $B - A$ would be the crescent of $B$ outside $A$, which is not shaded here.

Every symbol in this definition has a precise meaning:

• $A - B$ is read as “A minus B” — the order matters.
• $\{ x : \ldots \}$ is set-builder notation: “the set of all $x$ such that …”.
• $x \in A$ means $x$ is an element of $A$.
• $x \notin B$ means $x$ is not an element of $B$.

The figure also sets up an important observation that the textbook makes in the remark: the three sets $A - B$, $A \cap B$, and $B - A$ are mutually disjoint — no two of them share any element. You can see this in Fig 1.8 alone: the shaded region ($A - B$) does not touch the overlapping region ($A \cap B$), and the unshaded crescent of $B$ ($B - A$) is separate from both. Their intersections are all empty. …

Fig 1.9 is a Venn diagram that shows the three mutually disjoint pieces into which any two sets A and B partition the universal set U. The rectangle U contains two overlapping circles labelled A and B. The diagram is divided into three distinct regions, each marked with an arrow and a label:

• The left crescent (the part of A that does not overlap B) is labelled A − B.
• The central lens where the two circles overlap is labelled A ∩ B and is shaded with a deeper tone to distinguish it from the other two regions.
• The right crescent (the part of B that does not overlap A) is labelled B − A.

The key idea the figure teaches is that these three regions — A − B, A ∩ B, and B − A — are mutually disjoint. No two of them share any element. Their union is the entire set A ∪ B, and together with the region outside both circles (which is (A ∪ B)′), they form a partition of U.

The textbook uses this figure to reinforce the definition of set difference. For any two sets A and B:

The diagram makes it visually clear that A − B and B − A are generally different sets (unless A = B, in which case both are empty). It also shows why the remark in the textbook holds: the three pieces are disjoint because an element cannot simultaneously belong to A and not belong to A, nor can it be both in A ∩ B and outside the overlap.