Relations and Functions NCERT Solutions
NCERT Exercise 1.1
Question: 1. Determine whether each of the following relations are reflective, symmetric and transitive:
(i) Relation R in the set A = ( 1, 2, 3, --------, 13,14) defined as R = {(x, y) : 3x – y = 0 }
Solution:
Here A= { 1, 2, 3, ------,13, 14 }
R = {( x, y ) : 3x – y = 0 }
R = {( 1, 3 ), (2, 6), (3, 9), (4, 12)}
Now 1∈ A but (1,1)∉ R
So R is not reflexive relation.
(1,3) ∈R but (3, 1) ∉ R
So R is not symmetric relation.
(1, 3) ∈ R and (3, 9)∈R but (1,9 )∉R.
So R is not transitive relation.
Thus R is neither reflexive nor symmetric nor transitive.
(ii) Relation R in the set N of natural numbers define as R = {(x, y) : y = x + 5 and x < 4}
Solution:
Here R = {(x, y ): y = x + 5 and x < 4}
R = {(1, 6), (2, 7),(3, 8)}
A= {1, 2, 3}
Now 1 ∈ A but (1, 1) ∉ R
So R is not reflexive relation.
(1, 6)∈ R but (6,1) ∉ R.
So R is not symmetric relation.
In R, (a,b)∈ R but there is no ordered pair (b, c), ∈ R.
So R is not Transitive relation.
Thus R is neither reflexive nor symmetric nor transitive.
(iii) Relation R in the set A = {1, 2, 3, 4, 5,6} as R = {( x, y) : y is divisible by x }
Solution:
Here A ={1, 2, 3, 4, 5, 6}
R = {(x, y): ( x, y) ∈ A, y is divisible by x}
A = {(1, 1), (1, 2), (1, 3), (1, 4) (1, 5) (1, 6) (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (5, 5), (6, 6))}
Now 1, 2, 3, 4, 5, 6 ∈ A
There for (1,1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) ∈ R
So R is reflexive relation.
(1, 2 ) ∈ R but (2,1) ∉ R
So R is a not symmetric relation.
(1, 1) ∈R and (1, 2) ∈ R ⇒ (1, 2) ∈ R
(2, 2) ∈ R and (2, 4) ∈ R ⇒(2, 4) ∈ R
(3, 3) ∈ R and (3, 6) ∈ R ⇒(3,6) ∈R and so on.
So R is transitive relation.
Thus R is reflexive and transitive but not symmetric.
(iv) Relation R in the set Z of all integers defined as R = {(x, y): x – y is an integer}
Solution:
Here R = {(x, y): x – y is an integer, x ∈ Z,Y∈Z}
Now x – x = 0 is an integer ⇒ (x, x ) ∈ R.
So R is reflexive relation.
x – y is an integer than y – x is also an integer
(x, y) ∈ R ⇒(y, x) ∈ R.
So R is a symmetric relation.
x – y is an integer, y – z is an integer then x – z is also an integer.
Therefore, (x, y) ∈ R, (y, z ) ∈ R ⇒ (x, z)∈ R.
So R is a transitive relation.
Thus R is reflective, symmetric and transitive.
(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 }
Solution:
Here, given, R = {(x, y); x and y work at the same place}
Now place of work of both x and x is same
Therefore (x, x) ∈ R
So R is reflexive relation.
Place of work of x and y is same as place of work of y and x
There for (x, y) ∈ R = (y, x) ∈ R
So R is symmetric relation.
Place of work of x and y and place of work of y and z are same then place of work x and z is same.
Therefore (x, y) ∈R,(y, z) ∈ R ⇒ ( x, z)∈ R
So R is a transitive relation.
Thus R is reflexive, symmetric and transitive.
(v) Here A is a set of human beings in a town at a particular time.
(b) R = {( x, y) : x and y live in the same locality}
Solution:
Here, R = {(x, y ): x and y live in the same locality }
Now x and both live in the same locality
Therefore (x, x ) ∈ R
So R is reflective relation.
x and y live in the same locality then y and x also live in the same locality
Therefore ( x, y ) ∈ R = ( y, x ) ∈ R
So R is symmetric relation.
x and y live in the same locality then y and z also live in the same locality then x and z live n the same locality.
Therefore, ( x, y ) ∈ R, ( y, z ) ∈ R ⇒ ( x, z ) ∈ R
So, R is a transitive relation.
Thus, R is reflexive, symmetric and transitive.
(v) Relation R in the set A of human beings in a town at a particular time given by
(c)R = {(x, y) : x is exactly 7 cm taller than y }
Solution:
Here, given, R = {(x, y): x is exactly 7 cm taller than y}
Now x cannot be 7 cm taller than x.
So R is not reflexive relation.
If x is exactly 7 cm taller than y, then y cannot be taller than x.
Therefore (x, y) ∈ R ⇒ (y, x) ∉ R.
So R is not symmetric relation.
x is exactly 7 cm taller than y and y is exactly 7 cm taller than z then x cannot, be 7 cm taller than z.
So R is not transitive relation.
Thus R is neither reflexive nor transitive nor symmetric.
(v) Relation R in the set A of human beings in a town at a particular time given by
(d) R = {(x, y ) : x is wife of y }
Solution:
Here, given, R = {(x, y): x is wife of y}
Now x cannot be wife x.
So, R is not reflexive relation.
x is wife of y then y is not wife of x.
So R is not symmetric relation.
If x is wife of y but y cannot be wife of any body.
So R is not transitive relation.
Thus R is neither reflexive nor symmetric nor transitive.
(v) Relation R in the set A of human beings in a town at a particular time given by
(e) R = { ( x, y ) : x is father of y }
Solution:
Here, given, R = { ( x, y); x is a father of y }
Now x cannot be father of himself.
So R is not reflexive relation.
x is father of y than y cannot be father of x.
So R is not symmetric relation.
x is a father of y and y is a father of z then x cannot be father of z.
So R is not transitive relation.
Thus R is neither reflexive nor symmetric nor transitive.
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