Question

The relation RR defined on the set A =\{1,2,3,4,5\}A={1,2,3,4,5} by R =\left\{( a , b ):\left| a ^{2}- b ^{2}\right|< 1 6 ; a , b \in A \right\}R={(a,b):∣∣​a2−b2∣∣​<16;a,b∈A} is given by​

Answer 1

Topic - Set, Relation, Mapping

The given set is

A = {1, 2, 3, 4, 5}

The relation R is defined by

R = {(a, b) : | a² - b² | < 16; a, b ∈ A}

To check: if the relation R is an equivalence relation.

Reflexive relation:

• 1 ∈ A ⇒ |1² - 1²| = 0 < 16. 1R1 holds.
• 2 ∈ A ⇒ |2² - 2²| 0 < 16. 2R2 holds.
• 3 ∈ A ⇒ |3² - 3²| 0 < 16. 3R3 holds.
• 4 ∈ A ⇒ |4² - 4²| 0 < 16. 4R4 holds.
• 5 ∈ A ⇒ |5² - 5²| 0 < 16. 5R5 holds.

∴ R is reflexive.

Symmetric relation:

• (1, 2) ∈ A × A ⇒ |1² - 2²| = 3 < 16. 1R2 holds.
• (2, 1) ∈ A × A ⇒ |2² - 1²| = 3 < 16. 2R1 holds.

• (1, 3) ∈ A × A ⇒ |1² - 3²| = 8 < 16. 1R3 holds.
• (3, 1) ∈ A × A ⇒ |3² - 1²| = 8 < 16. 3R1 holds.

• (1, 4) ∈ A × A ⇒ |1² - 4²| = 15 < 16. 1R4 holds.
• (4, 1) ∈ A × A ⇒ |4² - 1²| = 15 < 16. 4R1 holds.

• (1, 5) ∈ A × A ⇒ |1² - 5²| = 24 > 16. 1R5 does not hold.

∵ 1R5 does not hold, the relation R cannot be defined on the set A.

∴ we conclude that R is neither a definable relation on the set A nor an equivalence relation.