Question

Show that the relation
R={(a,b):≤b^3} on the
Set of all real
R is neither reflexive
Symmetric
transitive.

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Answer 1

Answer:

let A = { set of all real numbers }

Reflexive: consider ordered pair (1/2, 1/2)

(1/2) not ≤ to (1/2)³ = 1/8

therefore (1/2,1/2) ∉ R

it is not reflexive

Symetric, consider ordered pair (1, 2)

1 ≤ 2³ => 1 ≤ 8 but 2 ≥ 1³

thetefore (1, 2)∈ R , (2, 1) ∉ R

hence it is not symetric

transitive

comsider ordered pairs (4 ,2) (2, 3/2)

4 ≤ 2² => 4 ≤ 4 , 2 ≤ (3/2)² => 4 ≤ 2.25

but 4 > (3/2)³ => 4 > 3.375

(4, 2) ∈ R, (2, 3/2) ∈ R (3, 3/2) ∉ R

it is not transitive

hence given relation is neither reflexive, symmetric not transitive