Sets — chapter notes

A set is a well-defined collection of objects; the objects are its elements. $a \in A$ means $a$ belongs to $A$.

• Roster form: list the elements once each, e.g. $\{7, 14, 21, 28, 35\}$.
• Set-builder form: give a rule, e.g. $\{x \in \mathbb{N} : x \text{ is a multiple of } 7,\ x < 40\}$.

Standard sets: $\mathbb{N}$, $\mathbb{W}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, each inside the next. For a set-builder rule, solve the condition inside the stated universe (negative integers count in $\mathbb{Z}$).

• Empty set $\varnothing$: no element. $\{\varnothing\}$ is not empty; it has one element.
• Singleton: exactly one element.
• Finite: counting ends; $n(A)$ is the number of elements.
• Infinite: counting never ends.
• Equal sets: exactly the same elements. Order and repeats do not matter. Equal counts do not make equal sets.

To show $A = B$, show $A \subseteq B$ and $B \subseteq A$.

$B \subseteq A$ if every element of $B$ is in $A$; $B \subset A$ if also $B \ne A$ (proper subset). $\varnothing \subseteq A$ and $A \subseteq A$ always.

The power set $P(A)$ is the set of all subsets. If $n(A) = n$: subsets $2^n$; proper subsets $2^n - 1$; non-empty proper subsets $2^n - 2$. Fixing $k$ elements inside leaves $2^{n-k}$ subsets.

The universal set $U$ contains every set of the discussion; it is the rectangle of a Venn diagram.

Intervals are subsets of $\mathbb{R}$. $(a, b)$ excludes both ends; $[a, b]$ includes both; $[a, b)$ and $(a, b]$ include one. $(a, \infty)$ and $(-\infty, b]$ are unbounded. $\infty$ is never included.

On the number line: filled circle = included, open circle = excluded.

Integers in an interval: count the end points carefully. $(-2, 3)$ has $4$, $[-2, 3]$ has $6$, $[-2, 3)$ has $5$.

• Union $A \cup B$: in $A$ or $B$ (or both).
• Intersection $A \cap B$: in both. Disjoint if $A \cap B = \varnothing$.
• Difference $A - B$: in $A$, not in $B$. In general $A - B \ne B - A$.
• Complement $A' = U - A$.

In a Venn diagram, label the regions: only $A$, both, only $B$, outside. For intervals, test every end point in both sets: $[-3, 2) \cap (0, 5] = (0, 2)$, $[-3, 2) \cup (0, 5] = [-3, 5]$.

• Commutative: $A \cup B = B \cup A$, $A \cap B = B \cap A$.
• Associative: $(A \cup B) \cup C = A \cup (B \cup C)$, and the same for $\cap$.
• Distributive: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$, $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.
• De Morgan: $(A \cup B)' = A' \cap B'$, $(A \cap B)' = A' \cup B'$.
• Identity / complement: $A \cup \varnothing = A$, $A \cap U = A$, $A \cup A' = U$, $A \cap A' = \varnothing$.

To verify a law, compute both sides for the given sets. To prove an identity, show each side is a subset of the other using a general element $x$.

Practical problems of this type are taught but not board assessed (formative assessment only).

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$, $n(A - B) = n(A) - n(A \cap B)$, $n(A') = n(U) - n(A)$. For disjoint sets, $n(A \cup B) = n(A) + n(B)$.

Three sets: $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$.

Method: draw a Venn diagram and fill the innermost region first, then work outwards. Check that the regions add up to $n(U)$.