Week 2: Sets and Relations

SETS & RELATIONS

The Universe of Sets

A set is a collection of distinct objects. But in mathematics, it’s the foundation of everything.

We typically use capital letters for sets and lowercase for elements.

A = 5

Element membership: $3 \in A$ (3 belongs to A)

\cup

Combining all elements from both sets.

\cap

Elements common to both sets.

-

Elements in one set but not the other.

A relation defines a connection between elements of two sets.

CONNECTING DOTS

💡 Cartesian Product

The Cartesian product of sets A and B, denoted $A \times B$ , is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$ .

$A \times B = \{(a, b) \mid a \in A, b \in B\}$

Reflexive: Every element is related to itself. $(a, a) \in R$ for all $a \in A$ .
Symmetric: If $a$ is related to $b$ , then $b$ is related to $a$ .
Transitive: If $a$ is related to $b$ and $b$ is related to $c$ , then $a$ is related to $c$ .

If a relation is Reflexive, Symmetric, AND Transitive, it is called an Equivalence Relation.

Important!