Relation Formulas for Class 11

Relation Formulas for Class 11

Here we are providing Relation Formula For Class 11 Maths. Practice Relation problems based on these formulas. Important terms and definitions are also included so that you can revise them in a very short time.

List of Relation Formulas for Class 11

(1) Ordered Pair: An ordered pair consists of two objects or elements grouped in a particular order.

Two numbers a and b listed in a specific order and enclosed in parentheses form an ordered pair (a, b) . Here a is the first component and b is the second component. In general, (a, b)≠(b, a) .

(2) Equality of Ordered Pairs: Two ordered pairs (a₁, b₁) and (a₂, b₂) are equal iff a₁=a₂ and b₁=b₂

(a, b) = (c, d) ⇔ a = c and b = d.

(3) Cartesian (or Cross) 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. A × B = {(a, b) : a ∈ A and b ∈ B}

If A = Φ  or B = Φ then A x B = Φ

(4) Ordered Triplet: Three numbers a, b, c listed in a specific order and enclosed in parentheses form an ordered triplet (a, b, c).

(a, b, c) ≠ (b, a, c) ≠ (c, a, b), etc.

A× B×C = {(a, b, c) : a ∈ A, b ∈ B and c ∈ C}

(5) For any nonempty sets A, B, C, we have
(A x B) x C = A x (B x C), each denoted by A x B x C.

(6) For any sets A, B and C, we have:

(i) A x (B ∪ C) = (A x B) ∪ (A x C)

(ii) A x (B ∩ C) = (A x B) ∩ (A x C)

(iii) A x (B – C) = (A x B) – (A x C)

(iv) (A x B) ∩ (B x A) = (A ∩ B) x (B ∩ A) = (A ∩ B) x (A ∩ B)

(v) A x B = A x C ⇒ B = C

(vi) A ⊂ B ⇒ (A x A) ⊂ (A x B) ∩ (B x A)

(vii) A ⊂ B ⇒ (A x C) ⊂ (B x C)

(viii) A ⊂ B and C ⊂ D ⇒ (A x C) ⊂ (B x D)

(ix) A x B = B x A ⇔ A = B

(7) Let A and B be two nonempty sets and let R ⊆ A x B. Then, R is called a relation from A to B.

>> If (a, b) ∈ R, we say that ‘a is related to b‘ and we write, a R b.

>> If (a, b) ∉ R, we say that ‘a is not related to b‘

>> Dom (R) = {a : (a, b) ∈ R}, range (R) = {b : (a, b) ∈ R} .

(8) We define, R^-1 = {(b, a) : (a, b) ∈ R},

dom (R) = range (R^-1) and range (R) = dom (R^-1).

(9) Let A be a nonempty set. Then, every subset of A x A is called a binary relation on A.

(10) If either A or B is an infinite set, then A × B is an infinite set.

(11) n(A × B) = n(A) × n(B)

(12) If A and B be any two non-empty sets having n elements in common, then A × B and B × A have n² elements in common.