Question

2. i) Let A = {1, 2, 3, … , 19, 20}, and R be the equivalence relation

on A defined by aRb if and only if a – b is divisible by 5. Find the

partition of A induced by R.​

Answer 1

Answer:

A relation is defined on Z such that aRb⇒(a−b) is divisible by 5,

For Reflexive: (a,a)∈R.

Since, (a−a)=0 is divisible by 5.  Therefore, the realtion is reflexive.

For symmetric: If (a,b)∈R⇒(b,a)∈R.

(a,b)∈R⇒(a−b) is divisible by 5.

Now, (b−a)=−(a−b) is also divisible by 5. Therefore, (b,a)∈R

Hence, the relation is symmetric.

For Transitive: If (a,b)∈R and (b,a)∈R⇒(a,c)∈R.

(a,b)∈R⇒(a−b) is divisible by 5.

(b,c)∈R⇒(b−c) is divisible by 5.

Then (a−c)=(a−b+b−c)=(a−b)+(b−c) is also divisible by 5.  Therefore, (a,c)∈R.

Hence, the relation is transitive.

There, the relation is equivalent.

Now, depending upon the remainder obtained when dividing (a−b) by 5 we can divide the set Z iinto 5 equivalent classes and they are disjoint i.e., there are no common elements between any two classes.

Step-by-step explanation:

Answer 2

Answer:

Step-by-step explanation:

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