Equivalence Relations
A relation $R$ on set $A$ is an Equivalence Relation if:
- Reflexive: $(a, a) \in R$ for all $a \in A$.
- Symmetric: $(a, b) \in R \implies (b, a) \in R$.
- Transitive: $(a, b) \in R$ and $(b, c) \in R \implies (a, c) \in R$.
Injective & Surjective Functions
- Injective (One-One): $f(x_1) = f(x_2) \implies x_1 = x_2$.
- Surjective (Onto): For every $y$ in codomain, there exists $x$ in domain such that $f(x) = y$ ($\text{Range} = \text{Codomain}$).
- Bijective: Both one-one and onto.
