Question

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 ): 3 x − 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 } (e) R = {( x , y ): x is father of y }

Answer 1

(i)

The given relation R in set A={1,2,3....,13,14} is defined as R={( x,y ):3x−y=0}.

Thus, R={ ( 1,3 ),( 2,6 ),( 3,9 ),( 4,12 ) }.

Since, ( 1,1 ),( 2,2 ),( 3,3 ).....( 14,14 )∉R hence R is not reflexive.

Since, ( 1,3 )∈R but ( 3,1 )∉R hence, Ris not symmetric.

Since, ( 1,3 ),( 3,9 )∈R but ( 1,9 )∉R, hence Ris not transitive.

Therefore, the given relation, R={ ( 1,3 ),( 2,6 ),( 3,9 ),( 4,12 ) } is neither reflexive, nor symmetric, nor transitive.

(ii)

The given relation Rin the set N of natural numbers is defined as R={ ( x,y ):y=x+5 and x<4 }

Thus, R={ ( 1,6 ),( 2,7 ),( 3,8 ) }.

Since, ( 1,1 ),( 2,2 ),( 3,3 ).....∉R hence Ris not reflexive.

Since, ( 1,6 )∈R but ( 6,1 )∉R hence, Ris not symmetric.

Since, there is no pair in R such that ( a 1 , a 2 ) and ( a 2 , a 3 )∈R then ( a 1 , a 3 )∉R, hence R is not transitive.

Therefore, the given relation R={ ( 1,6 ),( 2,7 ),( 3,8 ) } is neither reflexive, nor symmetric, nor transitive.

(iii)

The given relation R in the set A={ 1,2,3,4,5,6 } is defined as R={ ( x,y ):y is divisible by x }.

We know that, x is divisible by itself. Thus, ( 1,1 ),( 2,2 ),( 3,3 ).....∈R hence Ris reflexive.

Since, ( 2,4 )∈R as 4 is divisible by 2, but ( 4,2 )∉R as 2 is not divisible by 4. Hence, R is not symmetric.

Let ( x,y ),( y,z )∈R. Then, y is divisible by x and z is also divisible by x. Thus, ( x,z )∈R. Hence, R is transitive.

Therefore, the given relation R={ ( x,y ):y is divisible by x } in the set A={ 1,2,3,4,5,6 } is reflexive and transitive, but not symmetric.

(iv)

The given relation R in the set Z of all integers is defined as R={ ( x,y ):x−y is an integer }.

We know that for every number x∈Z, ( x,x )∈R, since, x−x=0 is an integer. Thus, ( 1,1 ),( 2,2 ),( 3,3 ).....∈R, hence R is reflexive.

Let ( x,y )∈R then x−y is an integer. Since, −( x−y ) is also an integer, thus ( y,x )∈R. Therefore, ( x,y ) and ( y,x )∈R. Hence, R is symmetric.

Let ( x,y ),( y,z )∈R. Then, ( x−y ) and ( y−z ) are integers.

( x−z )=( x−y )+( y−z ) is also an integer. Thus, ( x,z )∈R.

Therefore, ( x,y )∈R and ( y,z )∈R implies that ( x,z )∈R Hence, R is transitive.

Therefore, the given relation R={ ( x,y ):x−y is an integer } in the set Z of all integers is reflexive, symmetric and transitive.

(v)

(a)

The given relation R in set A of human beings in n a town at a particular time is defined as R={ ( x,y ):x and y work at the same place }.

( x,x )∈R, since, x and x both work at the same place. Thus, R is reflexive.

Let ( x,y )∈R then x and y work at the same place. Also ( y,x )∈R if x and y work at the same place, then y and x also work at the same place. Therefore, ( x,y ) and ( y,x )∈R. Hence, R is symmetric.

Let ( x,y ),( y,z )∈R. Then, x and y work at the same place and y and z also work at the same place. Therefore, x and z work at the same place. Thus, ( x,z )∈R.

Therefore, ( x,y )∈R and ( y,z )∈R imply that ( x,z )∈R. Hence, R is transitive.

Therefore, the given relation R={ ( x,y ):x and y work at the same place } in set A of human beings in a town at a particular time is reflexive, symmetric and transitive.

(b)

The given relation R in set A of human beings in n a town at a particular time is defined as R={ ( x,y ):x and y live in the same locality }.

( x,x )∈R, since, x and x both live in the same locality. Thus, R is reflexive.

Let ( x,y )∈R then x and y live in the same locality. Also ( y,x )∈R since if x and y live in the same locality, therefore y and x also live in the same locality. Therefore, ( x,y ) and ( y,x )∈R. Hence, R is symmetric.

Let ( x,y ),( y,z )∈R. Then, x and y live in the same locality and y and z also live in the same locality. Therefore, x and z live in the same locality. Thus, ( x,z )∈R.

Therefore, ( x,y )∈R and ( y,z )∈R imply that ( x,z )∈R Hence, R is transitive.

Therefore, the given relation R={ ( x,y ):x and y live in the same locality } in the set A of human beings in n a town at a particular time is reflexive, symmetric and transitive.

(c)

The given relation R in the set A of human beings in a town at a particular time is defined as R={ ( x,y ):x is exactly 7 cm taller than y  }.

( x,x )∉R, since, any human being, xcannot be 7 cm taller than himself. Thus, R is not reflexive.

Let ( x,y )∈R, then x is exactly 7 cm taller than y. Now, ( y,x )∉R if x is exactly 7 cm taller than y , and y cannot be exactly 7 cm taller than and x. Therefore, if ( x,y )∈R, then ( y,x )∉R. Hence, R is not symmetric.

Let ( x,y ),( y,z )∈R. Then, x is exactly 7 cm taller than y and y is exactly 7 cm taller than z. Now, x cannot be is exactly 7 cm taller than z. Thus, ( x,z )∉R.

Therefore, ( x,y )∈R and ( y,z )∈R but ( x,z )∉R. Hence, R is not transitive.

Therefore, the given relation R={ ( x,y ):x is exactly 7 cm taller than y  } in set A of human beings in a town at a particular time is not reflexive, symmetric or transitive.

(d)

The given relation R in set A of human beings in a town at a particular time is defined as R={ ( x,y ):x is wife of y  }.

( x,x )∉R, since, xcannot be wife of herself. Thus, R is not reflexive.

Let ( x,y )∈R, then x is wife of y. Now, ( y,x )∉R if x is wife of y, then y cannot be the wife of x. Therefore, if ( x,y )∈R, then ( y,x )∉R. Hence, R is not symmetric.

There is no pair in R such that ( x,y ) and ( y,z )∈R if x is wife of y, then y cannot be wife of z. Thus, ( x,z )∉R. Hence, R is not transitive.

Therefore, the given relation R={ ( x,y ):x is wife of y  } in set A of human beings in a town at a particular time is not reflexive, symmetric or transitive.

(e)

The given relation R in set A of human beings in a town at a particular time is defined as R={ ( x,y ):x is father of y  }.

( x,x )∉R, since, xcannot be father of himself, thus, R is not reflexive.

Let ( x,y )∈R, then x is father of y. Now, ( y,x )∉R if x is father of y, then y cannot be father of x. Therefore, if ( x,y )∈R, then ( y,x )∉R. Hence, R is not symmetric.

Let ( x,y ) and ( y,z )∈R, then x is father of y and y is father of z. But, ( x,z )∉R.Since x cannot be father of z. Hence, R is not transitive.

Therefore, the given relation R={ ( x,y ):x is father of y  } in set A of human beings in a town at a particular time is not reflexive, symmetric or transitive.