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Ordered Pair
An ordered pair consists of two objects or elements grouped in a particular order,
Equality of ordered Pairs
Two ordered pair (a1, b1) and (a2, b2) are equal iff a1 = a2 and b1 = b2
Cartesian Product of Sets
For two non-empty sets A and B, the set of all ordered pairs (a, b) such that a ∈ A and b ∈ B is called Cartesian Product A × B, i.e.
Suppose A = {a, b} and B = {p, q}
A × B = {a, b} × {p, q}
= {(a, p) (a, q) (b, p) (b, q)}
Properties of Cartesian Product
- n(A × B) = n(A) × n(B)
- A × (B ∪ C) = (A × B) ∪ (A × C)
- A × (B ∩ C) = (A × B) ∩ (A × C)
- A × (B – C) = (A × B) – (A × C)
- (A × B) ∩ (C × D) = (A ∩ B) × (C ∩ D)
- A × (B’ ∪ C’)’ = (A × B) ∩ (A × C)
- A × (B’ ∩ C’)’ = (A × B) ∪ (A × C)
Relation
If A and B are two non-empty sets, then a relation R from A to B is a subset of A × B.
If R ⊆ A × B and (a, b) ∈ R, then we say that a is related to b by the relation R, Written as aRb.
Representation of a relation
- Rooster form
- Builder form
- Arrow diagram
Domain, Codomain, and Range of a Relation
Domain: The set of all first components or coordinates of the ordered pairs belonging to R is called the domain of R.
Codomain: The set of all second components or coordinates of the ordered pairs belonging to R is called the range of R. Also, the set of B is called the codomain of relation R.
Thus, domain of R = {a: (a, b) ∈ R} and range of R = {b: (a, b) ∈ R}
Types of Relations
- Reflexive Relation: A relation R on a set A is said to be a reflexive relation if every element of A is related to itself. i.e. (a, a) ∈ R
- Symmetric Relation: A relation R on a set A is said to be a symmetric relation iff (a, b) ∈ R and (b, a) ∈ R.
- Transitive Relation: A relation R on a set A is said to be a transitive relation, Iff (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R.
- Equivalence Relation: A relation R on a set A is said to be an equivalence relation if it is simultaneously reflective, symmetric and transitive on A.
Inverse Relation
IF A and B are two non-empty sets and R be a relation from A to B, then the inverse of R, denoted by R-1, is a relation from B to A and is defined by R-1 = {(b, a) : (a, b) ∈ R}.
