14A

Sets and Venn diagrams

Set notation, equal sets, the size of a set, the empty set, union and intersection, subsets, complements, Venn diagrams and the counting rule.

The language of sets

A set is a collection of distinct objects, listed in braces: $A = \{2, 4, 6\}$. Order and repeats do not matter, so $\{1, 2\} = \{2, 1, 1\}$.
$|A|$ is the size of $A$ (the number of elements). The empty set $\varnothing$ has size $0$.
$A \cup B$ (union): in $A$ or $B$ or both. $A \cap B$ (intersection): in both. Sets with no common elements are disjoint.
$A \subseteq B$: every element of $A$ is in $B$. A set of $n$ elements has $2^n$ subsets, including $\varnothing$ and itself.
The universal set $U$ contains everything being considered. The complement $\overline{A}$ is everything in $U$ that is not in $A$.

Example 1●

$U = \{1, 2, \ldots, 10\}$, $A$ = the even numbers, $B$ = the multiples of $3$. Find $A \cap B$, $A \cup B$ and $|\overline{A}|$.

$A = \{2, 4, 6, 8, 10\}$ and $B = \{3, 6, 9\}$.

$A \cap B = \{6\}$ and $A \cup B = \{2, 3, 4, 6, 8, 9, 10\}$.

$\overline{A} = \{1, 3, 5, 7, 9\}$, so $|\overline{A}| = 5$.

Counting rule

$$|A \cup B| = |A| + |B| - |A \cap B|$$

Example 2●●

In a class of $30$, $18$ play soccer, $12$ play tennis and $5$ play both. How many play neither?

Both: $5$. Soccer only: $13$. Tennis only: $7$.

Fill a Venn diagram from the middle out.

At least one: $18 + 12 - 5 = 25$.

Neither: $30 - 25 = 5$.

Flashcards

$A cup B$

In $A$ or $B$ (or both)

$A cap B$

In both $A$ and $B$

$overline{A}$

Everything not in $A$

Counting rule

$|A \cup B| = |A| + |B| - |A \cap B|$

Subsets of an $n$-element set

$2^n$

Practice

With $A$ and $B$ from Example 1, find $|A \cap B|$.
Find $|A \cup B|$.
Find $|\overline{A}|$.
Two sets are disjoint when:
$|A| = 18$, $|B| = 12$ and $|A \cap B| = 5$. Find $|A \cup B|$.
Of $30$ students, $18$ play soccer, $12$ play tennis and $5$ play both. How many play neither?
The empty set is:
How many subsets does $\{a, b, c\}$ have?
$A \cap \overline{B}$ means the elements: