Sets

Describing Sets and Set Notation

A set is a well-defined collection of elements, meaning there is no ambiguity about whether an item belongs to it. For example, 'tall students' is not a set, but 'students over 170 cm' is.

• Listing elements: Members are written inside curly brackets: $V = \{a, e, i, o, u\}$.
• Defining by a rule: State a rule identifying the elements: $F = \{\text{positive factors of } 12\}$.
• Set equality: Two sets are equal if they have identical elements, regardless of order: $\{1, 2, 3\} = \{3, 1, 2\}$.
• The symbol $\in$ means 'is an element of' ($2 \in F$), while $\notin$ means 'is not an element of' ($5 \notin F$).
• The cardinal number, $\#A$, is the count of distinct elements in set $A$.
• The null set (empty set), $\emptyset$ or $\{\}$, contains no elements.
• A subset ($A \subseteq B$) means every element of $A$ is in $B$.

Set Operations and Brackets

The universal set, $U$, contains all elements under discussion. Four primary operations combine or modify sets:

Work out brackets first. With $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$, $A = \{1, 2, 3, 4\}$, $B = \{3, 4, 5\}$, and $C = \{4, 5, 6\}$:

• $(A \cap B) \cup C = \{3, 4\} \cup \{4, 5, 6\} = \{3, 4, 5, 6\}$
• $A \setminus (B \cup C) = \{1, 2, 3, 4\} \setminus \{3, 4, 5, 6\} = \{1, 2\}$
• $(A \cup B)' = \{6, 7, 8\}$, so $(A \cup B)' \cap C = \{6\}$

Standard Number Sets

Standard letters denote the core number sets:

• $\mathbb{N}$: Natural numbers (positive integers, $\{1, 2, 3, \dots\}$).
• $\mathbb{Z}$: Integers (positive and negative whole numbers, and zero, $\{\dots, -2, -1, 0, 1, 2, \dots\}$).
• $\mathbb{Q}$: Rational numbers (numbers writable as $\frac{a}{b}$, where $a, b \in \mathbb{Z}$ and $b \neq 0$).
• $\mathbb{R}$: Real numbers (all numbers on the number line).
• $\mathbb{R} \setminus \mathbb{Q}$: Irrational numbers (real numbers that cannot be written as fractions, such as $\sqrt{2}$ and $\pi$).

These sets nest inside one another: $\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R}$. For example, $-4 \in \mathbb{Z}$ and $-4 \in \mathbb{Q}$, but $-4 \notin \mathbb{N}$.

Commutative and Associative Properties

An operation is commutative if swapping the order gives the same result. An operation is associative if changing grouping brackets gives the same result.

Use a single counterexample to disprove a property. For $A = \{1, 2, 3\}$, $B = \{2, 3\}$, and $C = \{3\}$:

• Order matters: $A \setminus B = \{1\}$, but $B \setminus A = \emptyset$.
• Grouping matters: $(A \setminus B) \setminus C = \{1\} \setminus \{3\} = \{1\}$, but $A \setminus (B \setminus C) = A \setminus \{2\} = \{1, 3\}$.

Solving Venn Diagram Problems

A Venn diagram displays sets as circles inside a rectangle for $U$.

• Start with the central intersection of all three sets ($A \cap B \cap C$).
• Subtract the centre from inclusive intersection totals. Counts labelled 'only' already exclude the centre.
• Add the counts in every separate region, including the outside region, and set the total equal to $\#U$.

Key terms

Set

A well-defined collection of distinct elements.

Universal Set (U)

The complete set of all elements under consideration in a problem.

Cardinal Number (#A)

The total number of distinct elements contained in set A.

Null Set (∅)

A set that contains no elements, also written as {}.

Subset (⊆)

A relationship where every element of one set is also in another set.

Intersection (∩)

The set of elements shared in common between two or more sets.

Union (∪)

The set of all elements belonging to either set or to both sets.

Set Difference (\)

A \ B is the set of elements in A that are not in B.