Question 1. Let A be the set of all human beings in a town at a particular time. Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) R = {(x. y) x and y work at the same place}ย
(ii) R = {(x. y) x and y live in the same locality}
(iii) R = {(x. y) x is wife of y}ย
(iv) R = {(x. y) x is father of y}ย ย
Solution:
(i) Given the relation R = {(x, y): x and y work at the same place}
Now we need to check whether the relation is reflexive or not.ย
Check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ ย x โ A.
Let x be any element of R.
Then, x โ R ย
โ x and x work at the same place is true since they are same. ย ย ย ย
โ (x, x) โ R [condition for reflexive relation]
So, R is a reflexive relation.
Now we have to check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b) โ A.
Let (x, y) โ R
โ x and y work at the same place ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย [since it is given in the question]
โ y and x work at the same place ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย [same as โx and y work at the same placeโ]
โ (y, x) โ R
So, R is a symmetric relation also.
Now we have to check whether the given relation is Transitive relation or not. Relation R is said to be Transitive over set A if (x, y) โ R and (y, z) โ R then (x, z) โ ย R ย โ x, y, z โ A.
Let (x, y) โ R and (y, z) โ R.
Then, x and y work at the same place. ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย [Given]
y and z also work at the same place. ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย [(y, z) โ R]
โ x, y and z all work at the same place.
โ x and z work at the same place.
โ (x, z) โ R
Therefore, R is a transitive relation also.
So the relation R = {(x, y): x and y work at the same place} is a reflexive relation, symmetric relation and transitive relation as well.ย
(ii) Given the relation R = {(x, y): x and y live in the same locality}
Now we have to check whether the relation R is reflexive, symmetric and transitive or not.
Check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ ย x โ A.
Let x be any element of relation R.
Then, x โ R
It is given that x and x live in the same locality is true since they are the same.
So, R is a reflexive relation.
Now we have to check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b) โ A.
Let (x, y) โ R
โ x and y live in the same locality ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย [ it is given in the question ]
โ y and x live in the same locality ย ย ย ย ย ย [if x and y live in the same locality, then y and x also live in the same locality]
โ (y, x) โ R
So, R is a symmetric relation as well.
Now we have to check whether the given relation is Transitive relation or not. Relation R is said to be Transitive over set A if (x, y) โ R and (y, z) โ R then (x, z) โ ย R ย โ x, y, z โ A.
Let x, y and z be any elements of R and (x, y) โ R and (y, z) โ R.
Then,
x and y live in the same locality and y and z live in the same locality
โ x, y and z all live in the same locality
โ x and z live in the same locality
โ (x, z) โ R
So, R is a transitive relation also.
So the relation R = {(x, y): x and y live in the same locality} is a reflexive relation, symmetric relation and transitive relation as well.ย
(iii) Given R = {(x, y): x is wife of y}
Now we have to check whether the relation R is reflexive, symmetric and transitive relation or not.
Check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ ย x โ A.
Let x be an element of R.
Then, x is wife of x cannot be true. ย ย ย ย ย [since the same person cannot be the wife of herself]
โ (x, x) โ R
So, R is not a reflexive relation.
Check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b) โ A.
Let (x, y) โ R
โ x is wife of y
โ x is female and y is male
โ y cannot be wife of x as y is husband of x
โ (y, x) โ R ย
So, R is not a symmetric relation.
Check whether the given relation is Transitive relation or not. Relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ ย R ย โ x, y, z โ A.
Let (x, y) โ R, but (y, z) โ R
Since x is wife of y, but y cannot be the wife of z, since y is husband of x.
โ x is not the wife of z.
โ(x, z)โ R
So, R is a transitive relation.
Hence the given relation R = {(x, y): x is wife of y} is a transitive relation but not a reflexive and symmetric relation.ย
(iv) Given the relation R = {(x, y): x is father of y}
Now we have to check whether the relation R is reflexive, symmetric and transitive or not.
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ ย x โ A.
Let x be an arbitrary element of R.
Then, x is father of x cannot be true. ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย [since no one can be father of himself]
So, R is not a reflexive relation.
Check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
Let (x, y)โ R
โ x is the father of y.
โ y is son/daughter of x.
โ (y, x) โ R
So, R is not a symmetric relation.
Now, check whether the given relation is Transitive relation or not. Relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ ย R ย โ x, y, z โ A.
Let (x, y)โ R and (y, z)โ R.
Then, x is father of y and y is father of z
โ x is grandfather of z
โ (x, z)โ R
So, R is not a transitive relation.
Hence, the given relation R = {(x, y): x is father of y} is not a reflexive relation, not a symmetric relation and not a transitive relation as well.
ย
Question 2. Three relations R1, R2 and R3 are defined on a set A = {a, b, c} as follows:
R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
R2 = {(a, a)}
R3 = {(b, c)}
R4 = {(a, b), (b, c), (c, a)}.
Find whether or not each of the relations R1, R2, R3, R4 on A is (i) reflexive (ii) symmetric and (iii) transitive.
Solution:
i) Considering the relation R1, we have
R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
Now we have check R1 is reflexive, symmetric and transitive or not.
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ x โ A.
Given (a, a), (b, b) and (c, c) โ R1 ย ย ย ย [since each element maps to itself]
So, R1 is reflexive.
Check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
We see that for every ordered pair (x, y), there is a pair (y, x) present in the relation R1.
So, R1 is symmetric.
Transitive: A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ ย R ย โ ย x, y, z โ A.
In the relation, (a, b) โ R1, (b, c) โ R1 and also (a, c) โ R1
So, R1 is transitive.
Therefore, R1 is reflexive relation, symmetric relation and transitive relation as well.
(ii) Considering the relation R2, we have
R2 = {(a, a)}
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ x โ A.
We can see that (a, a) โ R2. ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย [since each element maps to itself]
So, R2 is a reflexive relation.
Check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
We can see that (a, a) โ R
โ (a, a) โ R.
So, R2 is symmetric.
Check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ R โ x, y, z โ A.
R2 is clearly a transitive relation. ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย [since there is only one element in it]
Therefore, R2 is reflexive relation, symmetric relation and transitive relation as well.
(iii) Considering the relation R3, we haveย
R3 = {(b, c)}
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ x โ A.
In the relation, (a, a)โ R3, (b, b)โ R3 neither (c, c) โ R3.ย
So, R3 is not reflexive. ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย [since all pairs of type (x, x) should be present in the relation]
Check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
In the relation, (b, c) โ R3, but (c, b) โ R3
So, R3 is not symmetric.
Check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ R โ x, y, z โ A.
R3 has only two elements.
Hence, R3 is transitive.
Therefore, R2 is transitive relation but not a reflexive relation and not a symmetric relation also.
(iv) Considering the relation R4, we have
R4 = {(a, b), (b, c), (c, a)}
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ x โ A.
In the relation, (a, a) โ R4, (b, b) โ R4 (c, c) โ R4
So, R4 is not a reflexive relation.
Check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
Here, (a, b) โ R4, but (b, a) โ R4.
So, R4 is not symmetric
Check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ R โ x, y, z โ A.
In the relation, (a, b) โ R4, (b, c) โ R4, but (a, c) โ R4
So, R4 is not a transitive relation.
Therefore, R2 is not a reflexive relation, not a symmetric relation and neither a transitive relation as well.
Question 3. Test whether the following relation R1, R2, and R3 are (i) reflexive (ii) symmetric and (iii) transitive:
(i) R1 on Q0 defined by (a, b) โ R1 โ a = 1/b.
(ii) R2 on Z defined by (a, b) โ R2 โ |a โ b| โค 5
(iii) R3 on R defined by (a, b) โ R3 โ a2 โ 4ab + 3b2 = 0.
Solution:
i) Given R1 on Q0 defined as (a, b) โ R1 โ a = 1/b.
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ ย x โ A.
Let a be an element of R1.
Then, a โ R1
โ a โ 1/a โ ย a โ Q0
So, R1 is not reflexive.
Now check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
Let (a, b) โ R1 ย
Then, (a, b) โ R1
Therefore, we can write โaโ as a =1/b
โ b = 1/a
โ (b, a) โ R1
So, R1 is symmetric.
Now check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ ย R ย โ ย x, y, z โ A.
Here, (a, b) โ R1 and (b, c) โ R2
โ a = 1/b and b = 1/c
โ a = 1/ (1/c) = c
โ a โ 1/c
โ (a, c) โ R1
So, R1 is not a transitive relation.
(ii) Given R2 on Z defined as (a, b) โ R2 โ |a โ b| โค 5
Now we have to check whether R2 is reflexive, symmetric and transitive or not.
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ ย x โ A.
Let a be an element of R2.
Then, a โ R2
On applying the given condition we will get,
โ | aโa | = 0 โค 5
So, R1 is a reflexive relation.
Now check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
Let (a, b) โ R2
โ |aโb| โค 5 ย ย ย ย ย ย ย ย ย ย [Since, |aโb| = |bโa|]
โ |bโa| โค 5
โ (b, a) โ R2
So, R2 is a symmetric relation.
Now check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ R โ x, y, z โ A.
Let (1, 3) โ R2 and (3, 7) โ R2
โ|1โ3|โค5 and |3โ7|โค5
But |1โ7| โฐ 5 ย
โ (1, 7) โ R2
So, R2 is not a transitive relation.
(iii) Given R3 on R defined as (a, b) โ R3 โ a2 โ 4ab + 3b2 = 0.
Now we have to check whether R2 is reflexive, symmetric and transitive or not.
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ x โ A.
Let a be an element of R3.
Then, a โ R3.
โ a2 โ 4a ร a+ 3a2= 0 ย
So, R3 is reflexive
Now check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
Let (a, b) โ R3
โ a2โ4ab+3b2=0
But b2โ4ba+3a2 โ 0 โ a, b โ R
So, R3 is not symmetric.
Now check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ R โ x, y, z โ A.
Let (1, 2) โ R3 and (2, 3) โ R3
โ 1 โ 8 + 6 = 0 and 4 โ 24 + 27 = 0
But 1 โ 12 + 9 โ 0
So, R3 is not a transitive relation.
Question 4. Let A = {1, 2, 3}, and let R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R2 = {(2, 2), (3, 1), (1, 3)}, R3 = {(1, 3), (3, 3)}. Find whether or not each of the relations R1, R2, R3 on A is (i) reflexive (ii) symmetric (iii) transitive.
Solution:
Considering the relation R1, we haveย
R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ x โ A.
Here, (1, 1) โ R, (2, 2) โ R, (3, 3) โ R
So, R1 is reflexive.
Check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
In the given relation, (2, 1) โ R1 but (1, 2) โ R1
So, R1 is not symmetric.
Now check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ ย R ย โ ย x, y, z โ A.
In the relation, (2, 1) โ R1 and (1, 3) โ R1 but (2, 3) โ R1 ย
So, R1 is not transitive.
Therefore, the relation R1 is reflexive but not symmetric and transitive relation.
Now considering the relation R2, we have
R2 = {(2, 2), (3, 1), (1, 3)}
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ x โ A.
Clearly, (1, 1) and (3, 3) โ R2 ย
So, R2 is not a reflexive relation.
Check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
In the relation, (1, 3) โ R2 and (3, 1) โ R2
So, R2 is a symmetric relation.
Now check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ R โ x, y, z โ A.
In the relation, (1, 3) โ R2 and (3, 1) โ R2 ย but (3, 3) โ R2
So, R2 is not a transitive relation.
Therefore, the relation R2 is symmetric but not a reflexive and transitive relation.
Considering the relation R3, we have
R3 = {(1, 3), (3, 3)}
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ ย x โ A.
In the relation, (1, 1) โ R3
So, R3 is not reflexive.
Check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
In the relation, (1, 3) โ R3, but (3, 1) โ R3
So, R3 is not symmetric.
Now check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ R โ x, y, z โ A.
Here, (1, 3) โ R3 and (3, 3) โ R3 ย
Also, (1, 3) โ R3
So, R3 is transitive.
Therefore, the relation R3 is transitive but not a reflexive and symmetric relation.
Question 5. The following relation is defined on the set of real numbers.
(i) aRb if a โ b > 0
(ii) aRb if 1 + a b > 0
(iii) aRb if |a| โค b.
Find whether relation is reflexive, symmetric or transitive.
Solution:
(i) Consider the relation defined as aRb if a โ b > 0
Now for this relation we have to check whether it is reflexive, transitive and symmetric or not.
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ x โ A.
Let a be an element of R.
Then, a โ R
But a โ a = 0 โฏ ย 0
So, this relation is not a reflexive relation.
Now check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
Let (a, b) โ R
โ a โ b > 0
โ โ (b โ a) > 0
โ b โ a < 0
So, the given relation is not a symmetric relation.
Now check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ ย R ย โ ย x, y, z โ A.
Let (a, b) โ R and (b, c) โ R. ย
Then, a โ b > 0 and b โ c > 0
Adding the two, we will get
a โ b + b โ c > 0
โ a โ c > 0 ย
โ (a, c) โ R.
So, the given relation is a transitive relation.
(ii) Consider the relation defined as aRb if (read as โif and only ifโ) 1 + a b > 0
Now for this relation we have to check whether it is reflexive, transitive and symmetric or not.
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ ย x โ A.
Let a be an element of R.
Then, a โ R
โ 1 + a ร a > 0
i.e. 1 + a2 > 0 ย ย ย ย ย ย [since, square of any number is positive]
So, the given relation is a reflexive relation.
Now check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
Let (a, b) โ R
โ 1 + a b > 0
โ 1 + b a > 0
โ (b, a) โ R
So, the given relation is symmetric.
Now check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ ย R ย โ ย x, y, z โ A.
Let (a, b) โ R and (b, c) โ R
โ1 + a b > 0 and 1 + b c >0
But 1+ ac โฏ ย 0
โ (a, c) โ R
So, the given relation is not a transitive relation.
(iii) Consider the relation defined as aRb if |a| โค b.
Now for this relation we have to check whether it is reflexive, transitive and symmetric or not.
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ ย x โ A.
Let a be an element of relation R.
Then, a โ R ย ย ย ย ย ย ย ย ย [Since, |a|=a]
โ |a| โฎ ย a
So, R is not a reflexive relation.
Now check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
Let (a, b) โ R
โ |a| โค b ย
โ |b| โฐ ย a โ a, b โ R
โ (b, a) โ R ย
So, R is not a symmetric relation.
Now check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ ย R ย โ ย x, y, z โ A.
Let (a, b) โ R and (b, c) โ R
โ |a| โค b and |b| โค c
Multiplying the corresponding sides, we will get
|a| ร |b| โค b c
โ |a| โค c
โ (a, c) โ R
Thus, R is a transitive relation.ย
Question 6. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Solution:
Given R = {(a, b): b = a + 1}
Now, for this relation we have to check whether it is reflexive, transitive and symmetric or not.ย
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ ย x โ A.
Let a be an element of R.
Then, a = a + 1 cannot be true for all a โ A.
โ (a, a) โ R ย
So, R is not a reflexive relation over the given set.
Now check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
Let (a, b) โ R
โ b = a + 1
โ โa = โb + 1
โ a = b โ 1
So, (b, a) โ R
Thus, R is not a symmetric relation over the given set.
Now check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ ย R ย โ ย x, y, z โ A.
Let (1, 2) and (2, 3) โ R
โ 2 = 1 + 1 and 3 ย
2 + 1 ย is true.
But 3 โ 1+1
โ (1, 3) โ ย R
So, R is not a transitive relation over the given set.
Question 7. Check whether the relation R on R defined as R = {(a, b): a โค b3} is reflexive, symmetric or transitive.
Solution:
We have given the relation R = {(a, b): a โค b3}
First let us check whether the given relation is reflexive or not.
It can be observed that (1/2, 1/2) in R as 1/2 > (1/2)3 = 1/8
So, R is not a reflexive relation.
Now, check for whether the relation is symmetric or not
(1, 2) โ R (as 1 < 23 = 8)
But,
(2, 1) โ R (as 2 > 13 = 1)
So, R is not a symmetric relation.
We have (3, 3/2), (3/2, 6/5) in โR asโ 3 < (3/2)3 and 3/2 < (6/5)3
But (3, 6/5) โ R as 3 > (6/5)3
So, R is not a transitive relation.
Hence, R is neither reflexive, nor symmetric, nor transitive.
Question 8. Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.
Solution:
We will verify this by taking example.
Let A be a set.
Then, Identity relation IA=IA is reflexive, since (a, a) โ A ย โ a โ A.
The converse of this need not be necessarily true.
Now, consider the set A = {1, 2, 3}
Here, relation R = {(1, 1), (2, 2) , (3, 3), (2, 1), (1, 3)} is reflexive on A.
But, R is not an identity relation.
Hence proved, that every identity relation on a set is reflexive but the converse is not necessarily true.
Question 9. If A = {1, 2, 3, 4} define relations on A which have properties of being
(i) Reflexive, transitive but not symmetric
(ii) Symmetric but neither reflexive nor transitive.
(iii) Reflexive, symmetric and transitive.ย
Solution:
(i) We have given the set A = {1, 2, 3, 4}
The relation on A having properties of being reflexive, transitive, but not symmetric is
R = {(1, 1), (2, 2), (3, 3), (4, 4), (2, 1)}
Relation R satisfies reflexivity and transitivity.
โ (1, 1), (2, 2), (3, 3) โ R ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย [satisfies the reflexivity property]
and (1, 1), (2, 1) โ R โ (1, 1) โ R ย ย ย ย ย ย ย ย ย ย ย ย ย [satisfies the transitivity property]ย
However, (2, 1) โ R, but (1, 2) โ R ย ย ย ย ย ย ย ย ย ย ย ย [does not satisfies the symmetric property]
(ii) We have given the set A = {1, 2, 3, 4}
The relation on A having properties of being reflexive, transitive, but not symmetric is
R = {(1, 1), (2, 2), (3, 3), (4, 4), (2, 1)}
Relation R satisfies reflexivity and transitivity.
โ (1, 1), (2, 2), (3, 3) โ R ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย [satisfies the reflexivity property]
And (1, 1), (2, 1) โ R โ (1, 1) โ R ย ย ย ย ย ย ย ย ย ย ย ย ย [satisfies the transitivity property]
However, (2, 1) โ R, but (1, 2) โ R ย ย ย ย ย ย ย ย ย ย ย ย [does not satisfies the symmetric property]
(iii) We have given the set A = {1, 2, 3, 4}
The relation on A having properties of being symmetric, reflexive and transitive is
R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)}
Relation R satisfies reflexivity, symmetricity and transitivity.
โ (1, 1), (2, 2), (3, 3) โ R ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย [satisfies the reflexivity property]
โ (1, 1) ย โ R and (2, 1) โ R ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย [satisfies the symmetric property]
โ (1, 1), (2, 1) โ R โ (1, 1) โ R ย ย ย ย ย ย ย ย ย ย ย ย ย ย [satisfies the transitivity property]
Question 10. Let R be a relation defined on the set of natural number N as R={(x, y): x, y โ N, 2x + y = 41}. Find the domain and range of R. Also verify whether R is (i) reflexive (ii) symmetric (iii) transitive.ย
Solution:
We have given,
{(x, y) : x, y โ N, 2x + y = 41}
Now,
2x + y = 41
โ y = 41 โ 2x
Put the value of x one by one to form the relation R.
The relation we will after putting x = 1, 2, 3, โฆโฆ. ,20 is:
[we canโt put x=21 since y = 41 โ 2(2) < 0, which is not a natural number] ยR = {(1, 39), (2, 37), (3, 35)โฆโฆโฆโฆ.., (20, 1)}
So the domain of R is
Domain(R) = {1, 2, 3, โฆโฆโฆ ,20}And the range of R is
Range(R) = {39, 37, 33, โฆโฆ. ,1} and can be rearranged as {1, 3, 5, โฆโฆโฆ.. ,39}ยNow for this relation we have to check whether it is reflexive, transitive and symmetric or not.
First let us check whether the relation is reflexive or not. A relation โRโ on a set โAโ is said to be reflexive if (x R x) โ x โ A i.e. (x, x) โ R โ ย x โ A.
Let x be an any element of relation R.
Since, (2, 2) โ R
So, R is not a reflexive relation.
Now check whether the relation is Symmetric relation or not. A relation R on set A is symmetric if (a, b)โ R and (b, a)โ R for all (a, b)โ A.
Since, (1, 39) โ R but (39, 1) โ R.
So, R is not symmetric.
Now check whether the relation is Transitive or not. A relation โRโ is said to be Transitive over set โAโ if (x, y) โ R and (y, z) โ R then (x, z) โ ย R ย โ ย x, y, z โ A.
Since, (15,11) โ R and (11,19) โ R but (15,19) โ R.
ย Thus, R is not transitive.