Mark the tick against the correct answer in the following:
Let Z be the set of all integers and let R be a relation on Z defined by a R b ⇔ a≥ b. Then, R is
According to the question ,
Given set Z = {1, 2, 3 ,4 …..}
And R = {(a, b) : a,b ∈ Z and a ≥ b}
Formula
For a relation R in set A
Reflexive
The relation is reflexive if (a , a) ∈ R for every a ∈ A
Symmetric
The relation is Symmetric if (a , b) ∈ R , then (b , a) ∈ R
Transitive
Relation is Transitive if (a , b) ∈ R & (b , c) ∈ R , then (a , c) ∈ R
Equivalence
If the relation is reflexive , symmetric and transitive , it is an equivalence relation.
Check for reflexive
Consider , (a,a) (b,b)
∴ a ≥ a and b ≥ b which is always true.
Therefore , R is reflexive ……. (1)
Check for symmetric
a R b ⇒ a ≥ b
b R a ⇒ b ≥ a
Both cannot be true.
Ex _ If a=2 and b=1
∴ 2 ≥ 1 is true but 1 ≥ 2 which is false.
Therefore , R is not symmetric ……. (2)
Check for transitive
a R b ⇒ a ≥ b
b R c ⇒ b ≥ c
∴ a ≥ c
Ex _a=5 , b=4 and c=2
∴ 5≥4 , 4≥2 and hence 5≥2
Therefore , R is transitive ……. (3)
Now , according to the equations (1) , (2) , (3)
Correct option will be (C)