Question 8

# Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = { (a,b) ; |a - b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Answer

*A* = {1, 2, 3, 4, 5}

R = { (a,b) ; |a – b| is even}

It is clear that for any element *a* ∈*A*, we have |a -a| = 0(which is even).

∴R is reflexive.

Let (*a*, *b*) ∈ R.

=> |a –b| is even.

=> |- (a –b)| = |b - a| is also even.

=> (b, a) ∈ R is even.

*A* = {1, 2, 3, 4, 5}

R = { (a, b) : | a – b| is even}

It is clear that for any element *a* ∈*A*, we have |a - a | = 0 (which is even).

∴R is reflexive.

Let (*a*, *b*) ∈ R.

⇒ |a –b| is even.

⇒ |- (a –b)| = |b - a| is also even.

⇒ (b, a) ∈ R is even.

∴R is symmetric.

Now, let (*a*, *b*) ∈ R and (*b*, *c*) ∈ R.

⇒ |a –b| is even and |(b –c)| is even.

⇒ (a – b) is even and (b –c ) is even.

⇒ (a –c ) = (a – b) + (b – c ) is even. [ Sum of two even integers is even]

⇒ |a – c | is even.

⇒ (*a*, *c*) ∈ R

∴R is transitive.

Hence, R is an equivalence relation.

Now, all elements of the set {1, 3, 5} are related to each other as all the elements of this subset are odd. Thus, the modulus of the difference between any two elements will be even.

Similarly, all elements of the set {2, 4} are related to each other as all the elements of this subset are even.

Also, no element of the subset {1, 3, 5} can be related to any element of {2, 4} as all elements of {1, 3, 5} are odd and all elements of {2, 4} are even. Thus, the modulus of the difference between the two elements (from each of these two subsets) will not be even.

∴R is symmetric.

Now, let (*a*, *b*) ∈ R and (*b*, *c*) ∈ R.

⇒ |a –b| is even and |(b –c)| is even.

⇒ (a – b) is even and (b –c ) is even.

⇒ (a –c ) = (a – b) + (b – c ) is even. [ Sum of two even integers is even]

⇒ |a – c | is even.

⇒ (*a*, *c*) ∈ R

∴R is transitive.

Hence, R is an equivalence relation.

Now, all elements of the set {1, 3, 5} are related to each other as all the elements of this subset are odd. Thus, the modulus of the difference between any two elements will be even.

Similarly, all elements of the set {2, 4} are related to each other as all the elements of this subset are even.

Also, no element of the subset {1, 3, 5} can be related to any element of {2, 4} as all elements of {1, 3, 5} are odd and all elements of {2, 4} are even. Thus, the modulus of the difference between the two elements (from each of these two subsets) will not be even.

- Q:- Given an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive. - Q:- Determine whether each of the following relations are reflexive, symmetric and transitive:

(i) Relation R in the set A = {1, 2, 3,13, 14} defined as

R = {(x, y): 3x − y = 0}

(ii) Relation R in the set N of natural numbers defined as

R = {(x, y): y = x + 5 and x < 4}

(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as

R = {(x, y): y is divisible by x}

(iv) Relation R in the set Z of all integers defined as

R = {(x, y): x − y is as integer}

(v) Relation R in the set A of human beings in a town at a particular time given by

(a) R = {(x, y): x and y work at the same place}

(b) R = {(x, y): x and y live in the same locality}

(c) R = {(x, y): x is exactly 7 cm taller than y}

(d) R = {(x, y): x is wife of y}

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(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

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- NCERT Chapter