Sets, Intervals and Venn Diagrams

A set is a collection of distinct objects, its elements. It can be given by listing, $\{1, 3, 5\}$, or by a rule (set-builder notation), $\{x \in \mathbb{R} : x > 2\}$.

• Two sets are equal when they have exactly the same elements.
• $A \subseteq B$ means every element of $A$ is also in $B$. Every set is a subset of itself, and $\varnothing$ is a subset of every set.
• A set with $n$ elements has $2^n$ subsets (each element is either in or out).

The table uses $A = \{1, 2, 3, 4\}$, $B = \{3, 4, 5\}$ and the universal set $U = \{1, 2, 3, 4, 5, 6\}$.

• $A$ and $B$ are disjoint when $A \cap B = \varnothing$.
• De Morgan's laws: $\overline{A \cup B} = \overline{A} \cap \overline{B}$ and $\overline{A \cap B} = \overline{A} \cup \overline{B}$.
• In the Cartesian product order matters: $(1, 5) \ne (5, 1)$, so $A \times B \ne B \times A$ in general, and $n(A \times B) = n(A) \cdot n(B)$.
• $A \times A$ is often written $A^2$; $\mathbb{R} \times \mathbb{R}$ is the coordinate plane.

• A square bracket includes the endpoint; a round bracket excludes it. Some Italian books write $]a, b[$ for $(a, b)$.
• $\pm\infty$ is never a number, so it always takes a round bracket.
• An answer such as $x < 1$ or $x > 3$ is a union of intervals: $(-\infty, 1) \cup (3, +\infty)$. Two conditions that must both hold give an intersection.
• $\mathbb{R} = (-\infty, +\infty)$ means every real number is a solution; $\varnothing$ means none is.

• Two sets: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.
• Elements in neither set: $n(U) - n(A \cup B)$.
• In A only: $n(A) - n(A \cap B)$.
• Three sets: add the three sizes, subtract the three pairwise overlaps, add back the triple overlap.
• With three sets, fill the Venn diagram from the centre outwards: the triple overlap first, then each pair minus the centre, then each set alone.