Let A ={ 1 , 2, 3, 4, 5,} , R ={ (a,b) such that ( a+ b) is even }, R is an equivalence relation on set A. Show that R is an equivalence relation.
Answers
Answer:
how to show that i dont know if u know pls tell me
Step-by-step explanation:
let A=1 and B 2 and C 3 and D 4
Therefore the relation is an equivalence relation.
Step-by-step explanation:
Given A={1,2,3,4,5} and R={(a,b): (a+b) is even}
Reflexive:
∀a ∈ A, (a,a)∈R Then R is reflexive.
(1,1),(2,2),(3,3),(4,4),(5,5)∈R
Since 1+1=2 which is a even number. Similarly for other.
Therefore R is reflexive.
Symmetric:
If (a,b)∈R , for ∀a,b∈ A⇒ (b,a)∈ R. Then R is symmetric.
(1,3) ∈ R [since 1+3= 4 which is a even number]
So, (3,1)∈ R since 1+3= 4 which is a even number.
Therefore R is symmetric.
Transitive:
(a,b),(b,c)∈ R, for ∀ a,b,c∈A ⇒ (a,c) ∈ R.
(1,3),(3,5)∈ R , [since 1+3= 4 and 3+5 =8 which are even numbers]
⇒(1,5)∈ R, since 1+5=6 is a even number.
Therefore R is transitive.
Since R is reflexive,symmetric and transitive.