Question

Example 20. Let I be the set of all integers, m be a positie integer and be the mean by
R = {(x, y): *.ye Ix-y is divisible by m). prove it​

Answer 1

Answer:

Sʜᴀʟʟ ᴡᴇ ʙᴇ ғʀɪᴇɴᴅs . ɪғ ʏᴏᴜ ᴡᴀɴɴᴀ ᴛʜᴇɴ

Step-by-step explanation:

Consider any a,b,c∈Z.

Since a−a=0=3.0⇒(a−a) is divisible by 3.

⇒(a,a)∈R⇒ is reflexive.

Let (a,b)∈R⇒(a−b) is divisible by 3.

⇒a−b=3q for some q∈Z⇒b−a=3(−q)

⇒(b−a) is divisible by 3 (∵q∈Z⇒−q∈Z⇒−q∈Z)

Thus, (a,b)∈R⇒(b,a)∈R⇒R is symmetric.

Let (a,b)∈R and (b,c)∈R

⇒(a−b) is divisible by 3 and (b−c) is divisible by 3

⇒a−b=3q and b−c=3q′ for some q,q′∈Z

⇒(a−b)+(b−c)=3(q+q′)⇒a−c=3(q+q′)

⇒(a−c) is divisible by 3 (∵q.q′∈Z⇒q+q′∈Z)

⇒(a,c)∈R

Thus, (a,b)∈R and (b,c)∈R⇒(a,c)∈R⇒R is transitive.

Therefore, the relation R is reflexive, symmetric and transitive, and hence it is an equivalence relation.