🔗 Relations and Functions

1. Cartesian Product of Sets

The Cartesian product A × B is the set of all ordered pairs (a, b) such that a ∈ A and b ∈ B.

Note: If n(A) = p and n(B) = q, then n(A × B) = pq.

2. Relations

A relation R from set A to set B is a subset of A × B. It is derived by describing a relationship between the first and second elements of ordered pairs in A × B.

🔗 Terms

Domain: The set of all first elements of the ordered pairs in a relation.

Range: The set of all second elements in a relation.

Codomain: The entire set B. (Note: Range ⊆ Codomain).

3. Functions

A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.

Vertical Line Test

A curve plotted on a graph represents a function if and only if no vertical line intersects the curve more than once.

4. Standard Real Functions

Identity Function: f(x) = x. (Line passing through origin).
Constant Function: f(x) = c. (Horizontal line).
Polynomial Function: f(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ.
Modulus Function: f(x) = |x|. V-shaped graph. Domain: R, Range: [0, ∞).
Signum Function: f(x) = 1 if x>0, 0 if x=0, -1 if x<0.
Greatest Integer Function: f(x) = [x], assumes the value of the greatest integer, less than or equal to x. Step-like graph.