Key Points

A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product $A \times B$. If $(a, b) \in R$, we say that 'a' is related to 'b' and write $a R b$.

An empty relation R in a set A is the relation where no element is related to any other, i.e., $R = \phi \subset A \times A$. A universal relation R is when every element of A is related to every element of A, i.e., $R = A \times A$.

A relation R on a set A is reflexive if $(a, a) \in R$ for every element $a \in A$. In simple terms, every element must be related to itself.

A relation R on a set A is symmetric if $(a_1, a_2) \in R$ implies that $(a_2, a_1) \in R$ for all $a_1, a_2 \in A$. If 'a' is related to 'b', then 'b' must be related to 'a'.

A relation R on a set A is transitive if $(a_1, a_2) \in R$ and $(a_2, a_3) \in R$ implies that $(a_1, a_3) \in R$ for all $a_1, a_2, a_3 \in A$.

A relation R on a set A is an equivalence relation if it is reflexive, symmetric, and transitive. This type of relation partitions the set into disjoint equivalence classes.

For an equivalence relation R on a set X, the equivalence class of an element $a \in X$, denoted by $[a]$, is the set of all elements in X that are related to a. That is, $[a] = \{x \in X : (x, a) \in R\}$.

A function $f$ from a set A to a set B, denoted $f: A \rightarrow B$, is a rule that assigns to each element $x \in A$ exactly one element $y \in B$. Set A is the domain and set B is the co-domain.

A function $f: X \rightarrow Y$ is one-one (or injective) if distinct elements in the domain have distinct images in the co-domain. To prove injectivity, show that $f(x_1) = f(x_2)$ implies $x_1 = x_2$.

A function $f: X \rightarrow Y$ is onto (or surjective) if every element in the co-domain Y has at least one pre-image in the domain X. To prove surjectivity, show that for any $y \in Y$, there exists an $x \in X$ such that $f(x) = y$.

A function is bijective if it is both one-one (injective) and onto (surjective). A bijective function creates a perfect one-to-one correspondence between the elements of its domain and co-domain.

Given two functions $f: A \rightarrow B$ and $g: B \rightarrow C$, their composition, denoted $g \circ f$, is a function from A to C defined by $(g \circ f)(x) = g(f(x))$ for all $x \in A$. Note that function composition is not generally commutative, i.e., $f \circ g \neq g \circ f$.

A function $f: X \rightarrow Y$ is invertible if there exists a function $g: Y \rightarrow X$ such that $g \circ f = I_X$ and $f \circ g = I_Y$, where $I_X$ and $I_Y$ are identity functions on sets X and Y respectively. The function g is called the inverse of f and is denoted by $f^{-1}$.

A function is invertible if and only if it is bijective (one-one and onto). This is the fundamental condition to check before attempting to find the inverse of a function.

For a function $f: X \rightarrow X$ where X is a finite set, the function is one-one if and only if it is onto. This is a special property that does not hold for infinite sets.