Question

3)Let A= {1,2,3,4), let R be the equivalence relation on AXA defined by (a, b) R
(c,d) if a +d = b +c. Find the equivalence class [(1,3)]​

Answer 1

Answer:

A={1,2,3...9}

R in A×A

(a,b) R (c,d) if (a,b)(c,d) ∈ A∈A

a+b=b+c

Consider (a,b) R (a,b) (a,b)∈A×A

a+b=b+a

Hence, R is reflexive.

Consider (a,b) R (c,d) given by (a,b) (c,d) ∈ A×A

a+d=b+c=>c+b=d+a

⇒(c,d)R(a,b)

Hence R is symmetric.

Let (a,b) R (c,d) and (c,d) R (e,f)

(a,b),(c,d),(e,f),∈A×A

a+b=b+c and c+f=d+e

a+b=b+c

⇒a−c=b−d-- (1)

c+f=d+e-- (2)

Adding (1) and (2)

a−c+c+f=b−d+d+e

a+f=b+e

(a,b)R(e,f)

R is transitive.

R is an equivalence relation.

We select from set A={1,2,3,....9}

a and b such that 2+b=5+a

so b=a+3

Consider (1,4)

(2,5) R (1,4)⇒2+4=5+1

[(2,5)=(1,4)(2,5),(3,6),(4,7),(5,8),(6,9)] is the equivalent class under relation R.