Question

Determine whether each of the following relations are reflexive, symmetric, and transitive.

1. Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
2. Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
3. Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
4. Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
5. Relation R in the set of human beings in a town at a particular time given by
   1. R={(x,y): x and y work at same place}
   2. R={(x,y): x and y live in the same locality}
   3. R={(x,y): x is exactly 7 am taller that y}
   4. R={(x,y): x is wife of y}
   5. R={(x,y):x is father of y}

Answer 1

(i) R={(1,3),(2,6),(3,3),(4,12)} 
R is not reflexive because (1,1)(2,2)...and(14,14)∉R Also, R is not symmetric as (1, 3) ∈R, but (3, 1) ∉ R. [3(3) − 1 ≠ 0] 
Also, R is not transitive as (1, 3), (3, 9) ∈R, but (1, 9) ∉ R. [3(1) − 9 ≠ 0] 
Hence, R is neither reflexive, nor symmetric, nor transitive. 

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(ii) R = {(x, y): y = x + 5 and x < 4} = {(1, 6), (2, 7), (3, 8)} It is seen that (1, 1) ∉ R. 
∴R is not reflexive. 
(1, 6) ∈R 
But,(1, 6) ∉ R. 
∴R is not symmetric. 
Now, since there is no pair in R such that (x, y) and (y, z) ∈ R so,we need to look for the ordered pair (x,y). therefore R is transitive.
Hence, R is neither reflexive ,nor symmetric but it is transitive.

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(iii) A = {1, 2, 3, 4, 5, 6} 
R = {(x, y): y is divisible by x} 
We know that any number (x) is divisible by itself. 
(x, x) ∈R 
∴R is reflexive. 
Now, (2, 4) ∈R [as 4 is divisible by 2] 
But, (4, 2) ∉ R. [as 2 is not divisible by 4] 
∴R is not symmetric. 
Let (x, y), (y, z) ∈ R. Then, y is divisible by x and z is divisible by y. 
∴z is divisible by x. 
⇒ (x, z) ∈R 
∴R is transitive. 
Hence, R is reflexive and transitive but not symmetric.

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(iv) R = {(x, y): x − y is an integer} 
Now, for every x ∈ Z, (x, x) ∈R as x − x = 0 is an integer. 
∴R is reflexive. 
Now, for every x, y ∈ Z if (x, y) ∈ R, then x − y is an integer. 
⇒ −(x − y) is also an integer.
⇒ (y − x) is an integer. 
∴ (y, x) ∈ R 
∴R is symmetric. 
Now, Let (x, y) and (y, z) ∈R, where x, y, z ∈ Z. 
⇒ (x − y) and (y − z) are integers. 
⇒ x − z = (x − y) + (y − z) is an integer ∴ (x, z) ∈R 
∴R is transitive. 
Hence, R is reflexive, symmetric, and transitive.

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(v) (a) R = {(x, y): x and y work at the same place} (x, x) ∈ R 
∴ R is reflexive. 
If (x, y) ∈ R, then x and y work at the same place. 
⇒ y and x work at the same place. 
⇒ (y, x) ∈ R. 
∴R is symmetric. 
Now, let (x, y), (y, z) ∈ R 
⇒ x and y work at the same place and y and z work at the same place. 
⇒ x and z work at the same place. 
⇒ (x, z) ∈R 
∴ R is transitive. 
Hence, R is reflexive, symmetric, and transitive.

(b) R = {(x, y): x and y live in the same locality}
Clearly (x, x) ∈ R as x and x is the same human being. 
∴ R is reflexive. 
If (x, y) ∈R, then x and y live in the same locality. 
⇒ y and x live in the same locality. 
⇒ (y, x) ∈ R 
∴R is symmetric. 
Now, let (x, y) ∈ R and (y, z) ∈ R. 
⇒ x and y live in the same locality and y and z live in the same locality. 
⇒ x and z live in the same locality. 
⇒ (x, z) ∈ R 
∴ R is transitive. 
Hence, R is reflexive, symmetric, and transitive.

(c) R = {(x, y): x is exactly 7 cm taller than y}
Now, (x, x) ∉ R 
Since human being x cannot be taller than himself.
∴R is not reflexive. 
Now, let (x, y) ∈R. 
⇒ x is exactly 7 cm taller than y. 
Then, y is not taller than x. 
∴ (y, x) ∉R 
Indeed if x is exactly 7 cm taller than y, then y is exactly 7 cm shorter than x. 
∴R is not symmetric. 
Now, Let (x, y), (y, z) ∈ R. 
⇒ x is exactly 7 cm taller than y and y is exactly 7 cm taller than z.
 ⇒ x is exactly 14 cm taller than z . 
∴(x, z) ∉R 
∴ R is not transitive. 
Hence, R is neither reflexive, nor symmetric, nor transitive.

(d) R = {(x, y): x is the wife of y}
Now, (x, x) ∉ R 
Since x cannot be the wife of herself. 
∴R is not reflexive. 
Now, let (x, y) ∈ R 
⇒ x is the wife of y. 
Clearly y is not the wife of x. 
∴(y, x) ∉ R 
Indeed if x is the wife of y, then y is the husband of x. 
∴ R is not transitive. 
Let (x, y), (y, z) ∈ R 
⇒ x is the wife of y and y is the wife of z. 
This case is not possible. Also, this does not imply that x is the wife of z. 
∴(x, z) ∉ R 
∴R is not transitive. 
Hence, R is neither reflexive, nor symmetric, nor transitive.

(e) R = {(x, y): x is the father of y}
(x, x) ∉ R 
As x cannot be the father of himself. 
∴R is not reflexive. 
Now, let (x, y) ∈R. 
⇒ x is the father of y. 
⇒ y cannot be the father of y. 
Indeed, y is the son or the daughter of y. 
∴(y, x) ∉ R 
∴ R is not symmetric. 
Now, let (x, y) ∈ R and (y, z) ∈ R. 
⇒ x is the father of y and y is the father of z. 
⇒ x is not the father of z. 
Indeed x is the grandfather of z. 
∴ (x, z) ∉ R 
∴R is not transitive. 
Hence, R is neither reflexive, nor symmetric, nor transitive.