Intersection of Sets

The intersection of two sets $A$ and $B$ is the set containing every element that is present in both $A$ and $B$ at the same time. Symbolically, $A \cap B = \{ x : x \in A \text{ and } x \in B \}$.

Intuition: It's the "common ground" — only the items shared by both groups. …

Fig. 1.5 is a Venn diagram inside a rectangle labelled $U$ (the universal set). Two overlapping circles are drawn: the left circle is labelled $A$, the right circle is labelled $B$. The region where the two circles overlap — the lens-shaped area common to both $A$ and $B$ — is shaded. An arrow points to this shaded region, and the label next to it is $A \cap B$.

The diagram teaches the idea of intersection: the set of all elements that belong to both $A$ and $B$ at the same time. The shaded overlap is the visual representation of that common ground. If an element lies in the shaded region, it must be inside circle $A$ and inside circle $B$. Any element that lies only in $A$ (the left crescent) or only in $B$ (the right crescent) is not part of the intersection.

The central formula the textbook develops with this figure is:

Here, the symbol $\cap$ is read as "intersection". The set-builder notation $\{ x : \dots \}$ means "the set of all $x$ such that …". The condition $x \in A$ and $x \in B$ forces $x$ to be in both sets simultaneously. The figure makes this abstract condition concrete: the shaded region is exactly where the membership conditions $x \in A$ and $x \in B$ are both true. …

Fig 1.6 is a Venn diagram that shows two disjoint sets. A rectangle labelled $U$ (the universal set) encloses two separate circles, one labelled $A$ and the other labelled $B$. The circles do not touch or overlap — there is a clear gap between them. No region belongs to both circles simultaneously.

The physical idea is simple: two sets that share no common element. The diagram makes the definition of disjoint sets visually immediate. Where Fig 1.5 (the previous figure in the textbook) shades the overlapping region of two intersecting circles, Fig 1.6 shows the opposite case — the intersection region is empty.

The key formula that this figure illustrates is the definition of disjoint sets:

Here $A$ and $B$ are any two sets, $\cap$ is the intersection operator (read as "cap" or "intersection"), and $\phi$ (the empty set symbol) denotes the set with no elements. The equation says: the set of elements common to both $A$ and $B$ is empty — there are none.

The textbook uses this figure immediately after giving the numerical example $A = \{2,4,6,8\}$ and $B = \{1,3,5,7\}$. Those two sets have no element in common, so their Venn diagram looks exactly like Fig 1.6.

The five Venn diagrams in Fig 1.7 are a visual proof of the distributive law for sets:

Each panel shows the same universal set $U$ (a rectangle) containing three overlapping circles labelled $A$ (upper-left), $B$ (upper-right), and $C$ (bottom). The shaded region in each panel represents the result of a set operation. By comparing the shaded areas in panels (ii) and (v), you see they are identical — which is exactly why the two sides of the equation are equal.

Panel (i): $B \cup C$ — the entire region belonging to either $B$ or $C$ (or both) is shaded. This is the union of the two circles on the right and bottom.

Panel (ii): $A \cap (B \cup C)$ — only the parts of $A$ that also lie inside $B$ or $C$ are shaded. In other words, take the shaded area from panel (i) and keep only the portion that falls inside circle $A$.

Panel (iii): $A \cap B$ — the overlap of circles $A$ and $B$ is shaded. This is the region common to both.

Panel (iv): $A \cap C$ — the overlap of circles $A$ and $C$ is shaded.

Panel (v): $(A \cap B) \cup (A \cap C)$ — the union of the two shaded regions from panels (iii) and (iv) is shaded. This is the combined area where $A$ overlaps with either $B$ or $C$.

The figure is placed in the textbook right after the statement of the distributive law (property v of intersection). It is not a proof in the formal sense, but a geometric intuition: if you shade the region described by the left-hand side, then shade the region described by the right-hand side, you get the same picture. That is why the law holds.