Revision: Mathematics >> Relations and Functions CUET (UG) Relations and Functions

A relation from set A to set B is any subset of the Cartesian product \[A \times B\].

So, if \[R \subseteq A \times B\], then R is a relation from A to B.

• Domain: The set of all first components of the ordered pairs in a relation R is called the domain of the relation R.

• Codomain: If R is a relation from A to B, then the set B is called the co–domain of the relation R.

• Range: The set of all second components of all ordered pairs in a relation R is called the range of the relation.

A function from set A to set B is a relation in which every element of A has exactly one image in B.

Condition for a Function

• every element of the domain must be used

• no element of the domain can have more than one image

• different elements may have the same image

An ordered pair is a pair of objects whose components occur in a special order. It is written by listing the two components in the specified order, separating them by a comma and enclosing the pair in parentheses. In the ordered pair (a, b), a is called the first component and b the second component.

• Two ordered pairs are equal only if their corresponding components are equal, so (a,b) = (c,d) if and only if a = c and b = d.

• In general, (a,b) ≠ (b,a)(a,b).

If A and B are two non-empty sets, then their Cartesian product is written as \[A \times B\].

• It is the set of all ordered pairs \[(a, b)\] such that \[a \in A\] and \[b \in B\].

• \[A \times B = \{(a, b) : a \in A, b \in B\}\].

• Range: Set of actual output values of f
• Range ⊆ Codomain

f: X → Y is a function if each element of X is associated with a unique element of Y

• Domain (X): Set of all input values
• Codomain (Y): Set of all possible outputs

A relation R on set A is said to be an equivalence relation if
(i) it is reflexive,
(ii) it is symmetric and
(iii) it is transitive.

Relation R on set A satisfying all the above three properties is an equivalence relation.

A relation R on a set A is said to be transitive if

(a, b) ∈ R and (b, c) ∈ R

⇒ (a, c) ∈ R for all a, b , c ∈ R

i.e. aRb and bRc

⇒ aRc for all a, b, c ∈ R

A relation R on a set A is said to be symmetric if

(a, b) ∈ R

⇒  (b, a)  ∈ R for all a b ∈ A

i.e.  aRb ⇒ bRa for all a , b ∈ A

• Empty relation and universal relation are called trivial relations.

• Identity relation always contains all self-pairs and is always reflexive.

• To check reflexivity, see whether every (a,a) is present in the relation.

• To check symmetry, see whether every (a,b) has the reverse pair (b,a).

• To check transitivity, see whether (a,b) and (b,c) together give (a,c).

• A relation is an equivalence relation only when all three properties—reflexive, symmetric, and transitive—are satisfied together.