RD Sharma Class 12 Solutions - Relations | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


1. Relations
Exercise 1.1
1 A. Question
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:
R = {(x, y) : x and y work at the same place}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x and y work at the same place}
Check for Reflexivity:
Since x & x are the same people then, x & x works at the same place.
Take yourself, for example, if you work at Bloomingdale then you work at Bloomingdale.
Since you can’t work in two places at a particular time,
So, ? x ? A, then (x, x) ? R.
? R is Reflexive.
Check for Symmetry:
If x & y works at the same place, then, y and x also work at the same place.
If you & your friend, Chris was working in Bloomindale, then Chris and you are working in Bloomingdale only.
The only difference is in the way of writing, either you write your name and your friend’s name or your
friend’s name and your name, it’s the same.
So, if (x, y) ? R, then (y, x) ? R
? x, y ? A
? R is Symmetric.
Check for Transitivity:
If x & y works at the same place and y & z works at the same place.
Then, x & z also works at the same place.
Say, if she & I was working in Bloomingdale and she & you were also working in Bloomingdale. Then, you and
I are working in the same company.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
? x, y, z ? A
Page 2


1. Relations
Exercise 1.1
1 A. Question
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:
R = {(x, y) : x and y work at the same place}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x and y work at the same place}
Check for Reflexivity:
Since x & x are the same people then, x & x works at the same place.
Take yourself, for example, if you work at Bloomingdale then you work at Bloomingdale.
Since you can’t work in two places at a particular time,
So, ? x ? A, then (x, x) ? R.
? R is Reflexive.
Check for Symmetry:
If x & y works at the same place, then, y and x also work at the same place.
If you & your friend, Chris was working in Bloomindale, then Chris and you are working in Bloomingdale only.
The only difference is in the way of writing, either you write your name and your friend’s name or your
friend’s name and your name, it’s the same.
So, if (x, y) ? R, then (y, x) ? R
? x, y ? A
? R is Symmetric.
Check for Transitivity:
If x & y works at the same place and y & z works at the same place.
Then, x & z also works at the same place.
Say, if she & I was working in Bloomingdale and she & you were also working in Bloomingdale. Then, you and
I are working in the same company.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
? x, y, z ? A
? R is Transitive.
Hence, R is reflexive, symmetric and transitive.
1 B. Question
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:
R = {(x, y) : x and y live in the same locality}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x and y live in the same locality}
Check for Reflexivity:
Since x & x are the same people, then, x & x live in the same locality.
Take yourself, for example, if you lived in colony x then you live in colony x.
Since you can’t live in two places at a particular time.
So, ? x ? A, then (x, x) ? R.
? R is Reflexive.
Check for Symmetry:
If x & y live in the same locality, then, y & x also lives in the the same locality.
If you & your friend, Chris are neighbors, then you and Chris are neighbors only.
The only difference is in the way of writing, either you write your name and your friend’s name or your
friend’s name and your name, it’s the same.
So, if (x, y) ? R, then (y, x) ? R.
? x, y ? R
? R is Symmetric.
Check for Transitivity:
If x & y lives in the same locality and y & z lives in the same locality.
Then, x & z also lives in the same locality.
Say, if she and I were living in colony x and she & you were also working in colony x. Then, you and I are
living in the same colony.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
? x, y, z ? A
? R is Transitive.
Page 3


1. Relations
Exercise 1.1
1 A. Question
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:
R = {(x, y) : x and y work at the same place}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x and y work at the same place}
Check for Reflexivity:
Since x & x are the same people then, x & x works at the same place.
Take yourself, for example, if you work at Bloomingdale then you work at Bloomingdale.
Since you can’t work in two places at a particular time,
So, ? x ? A, then (x, x) ? R.
? R is Reflexive.
Check for Symmetry:
If x & y works at the same place, then, y and x also work at the same place.
If you & your friend, Chris was working in Bloomindale, then Chris and you are working in Bloomingdale only.
The only difference is in the way of writing, either you write your name and your friend’s name or your
friend’s name and your name, it’s the same.
So, if (x, y) ? R, then (y, x) ? R
? x, y ? A
? R is Symmetric.
Check for Transitivity:
If x & y works at the same place and y & z works at the same place.
Then, x & z also works at the same place.
Say, if she & I was working in Bloomingdale and she & you were also working in Bloomingdale. Then, you and
I are working in the same company.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
? x, y, z ? A
? R is Transitive.
Hence, R is reflexive, symmetric and transitive.
1 B. Question
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:
R = {(x, y) : x and y live in the same locality}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x and y live in the same locality}
Check for Reflexivity:
Since x & x are the same people, then, x & x live in the same locality.
Take yourself, for example, if you lived in colony x then you live in colony x.
Since you can’t live in two places at a particular time.
So, ? x ? A, then (x, x) ? R.
? R is Reflexive.
Check for Symmetry:
If x & y live in the same locality, then, y & x also lives in the the same locality.
If you & your friend, Chris are neighbors, then you and Chris are neighbors only.
The only difference is in the way of writing, either you write your name and your friend’s name or your
friend’s name and your name, it’s the same.
So, if (x, y) ? R, then (y, x) ? R.
? x, y ? R
? R is Symmetric.
Check for Transitivity:
If x & y lives in the same locality and y & z lives in the same locality.
Then, x & z also lives in the same locality.
Say, if she and I were living in colony x and she & you were also working in colony x. Then, you and I are
living in the same colony.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
? x, y, z ? A
? R is Transitive.
Hence, R is reflexive, symmetric and transitive.
1 C. Question
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:
R = {(x, y) : x is wife of y}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x is wife of y}
Check for Reflexivity:
Since x and x are the same people.
Then, x cannot be the wife of itself.
A person cannot be a wife of itself.
Wendy is the wife of Sam; Wendy can’t be the wife of herself.
So, ? x ? A, then (x, x) ? R.
? R is not reflexive.
Check for Symmetry:
If x is the wife of y.Then, y cannot be the wife of x.
If Wendy is the wife of Sam, then Sam is the husband of Wendy.
Sam cannot be the wife of Wendy.
So, if (x, y) ? R, then (y, x) ? R.
? x, y ? A
? R is not symmetric.
Check for Transitivity:
If x is the wife of y and y is the wife of z, which is not logically possible.
Then, x is not the wife of z.
It’s easy, take Wendy, Sam, and Mac.
If Wendy is the wife of Sam, Sam can’t be the wife of Mac.
Thus, the possibility of Wendy being the wife of Mac also eliminates.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
? x, y, z ? A
Page 4


1. Relations
Exercise 1.1
1 A. Question
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:
R = {(x, y) : x and y work at the same place}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x and y work at the same place}
Check for Reflexivity:
Since x & x are the same people then, x & x works at the same place.
Take yourself, for example, if you work at Bloomingdale then you work at Bloomingdale.
Since you can’t work in two places at a particular time,
So, ? x ? A, then (x, x) ? R.
? R is Reflexive.
Check for Symmetry:
If x & y works at the same place, then, y and x also work at the same place.
If you & your friend, Chris was working in Bloomindale, then Chris and you are working in Bloomingdale only.
The only difference is in the way of writing, either you write your name and your friend’s name or your
friend’s name and your name, it’s the same.
So, if (x, y) ? R, then (y, x) ? R
? x, y ? A
? R is Symmetric.
Check for Transitivity:
If x & y works at the same place and y & z works at the same place.
Then, x & z also works at the same place.
Say, if she & I was working in Bloomingdale and she & you were also working in Bloomingdale. Then, you and
I are working in the same company.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
? x, y, z ? A
? R is Transitive.
Hence, R is reflexive, symmetric and transitive.
1 B. Question
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:
R = {(x, y) : x and y live in the same locality}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x and y live in the same locality}
Check for Reflexivity:
Since x & x are the same people, then, x & x live in the same locality.
Take yourself, for example, if you lived in colony x then you live in colony x.
Since you can’t live in two places at a particular time.
So, ? x ? A, then (x, x) ? R.
? R is Reflexive.
Check for Symmetry:
If x & y live in the same locality, then, y & x also lives in the the same locality.
If you & your friend, Chris are neighbors, then you and Chris are neighbors only.
The only difference is in the way of writing, either you write your name and your friend’s name or your
friend’s name and your name, it’s the same.
So, if (x, y) ? R, then (y, x) ? R.
? x, y ? R
? R is Symmetric.
Check for Transitivity:
If x & y lives in the same locality and y & z lives in the same locality.
Then, x & z also lives in the same locality.
Say, if she and I were living in colony x and she & you were also working in colony x. Then, you and I are
living in the same colony.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
? x, y, z ? A
? R is Transitive.
Hence, R is reflexive, symmetric and transitive.
1 C. Question
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:
R = {(x, y) : x is wife of y}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x is wife of y}
Check for Reflexivity:
Since x and x are the same people.
Then, x cannot be the wife of itself.
A person cannot be a wife of itself.
Wendy is the wife of Sam; Wendy can’t be the wife of herself.
So, ? x ? A, then (x, x) ? R.
? R is not reflexive.
Check for Symmetry:
If x is the wife of y.Then, y cannot be the wife of x.
If Wendy is the wife of Sam, then Sam is the husband of Wendy.
Sam cannot be the wife of Wendy.
So, if (x, y) ? R, then (y, x) ? R.
? x, y ? A
? R is not symmetric.
Check for Transitivity:
If x is the wife of y and y is the wife of z, which is not logically possible.
Then, x is not the wife of z.
It’s easy, take Wendy, Sam, and Mac.
If Wendy is the wife of Sam, Sam can’t be the wife of Mac.
Thus, the possibility of Wendy being the wife of Mac also eliminates.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
? x, y, z ? A
? R is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive.
1 D. Question
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:
R = {(x, y) : x is father of y}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x is father of y}
Check for Reflexivity:
Since x and x are the same people.
Then, x cannot be the father of itself.
A person cannot be a father of itself.
Leo is the father of Thiago
So, ? x ? A, then (x, x) ? R.
? R is not reflexive.
Check for Symmetry:
If x is the father of y.
Then, y cannot be the father of x.
If Sam is the father of Mac, then Mac is the son of Sam.
Mac cannot be the father of Sam.
So, if (x, y) ? R, then (y, x) ? R.
? x, y ? A
? R is not symmetric.
Check for Transitivity:
If x is the father of y and y is the father of z, then, x is not the father of z.
Take Mickey, Sam, and Mac.
If Mickey is the father of Sam, and Sam is the father of Mac.
Thus, Mickey is not the father of Mac, but the grandfather of Mac.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
Page 5


1. Relations
Exercise 1.1
1 A. Question
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:
R = {(x, y) : x and y work at the same place}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x and y work at the same place}
Check for Reflexivity:
Since x & x are the same people then, x & x works at the same place.
Take yourself, for example, if you work at Bloomingdale then you work at Bloomingdale.
Since you can’t work in two places at a particular time,
So, ? x ? A, then (x, x) ? R.
? R is Reflexive.
Check for Symmetry:
If x & y works at the same place, then, y and x also work at the same place.
If you & your friend, Chris was working in Bloomindale, then Chris and you are working in Bloomingdale only.
The only difference is in the way of writing, either you write your name and your friend’s name or your
friend’s name and your name, it’s the same.
So, if (x, y) ? R, then (y, x) ? R
? x, y ? A
? R is Symmetric.
Check for Transitivity:
If x & y works at the same place and y & z works at the same place.
Then, x & z also works at the same place.
Say, if she & I was working in Bloomingdale and she & you were also working in Bloomingdale. Then, you and
I are working in the same company.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
? x, y, z ? A
? R is Transitive.
Hence, R is reflexive, symmetric and transitive.
1 B. Question
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:
R = {(x, y) : x and y live in the same locality}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x and y live in the same locality}
Check for Reflexivity:
Since x & x are the same people, then, x & x live in the same locality.
Take yourself, for example, if you lived in colony x then you live in colony x.
Since you can’t live in two places at a particular time.
So, ? x ? A, then (x, x) ? R.
? R is Reflexive.
Check for Symmetry:
If x & y live in the same locality, then, y & x also lives in the the same locality.
If you & your friend, Chris are neighbors, then you and Chris are neighbors only.
The only difference is in the way of writing, either you write your name and your friend’s name or your
friend’s name and your name, it’s the same.
So, if (x, y) ? R, then (y, x) ? R.
? x, y ? R
? R is Symmetric.
Check for Transitivity:
If x & y lives in the same locality and y & z lives in the same locality.
Then, x & z also lives in the same locality.
Say, if she and I were living in colony x and she & you were also working in colony x. Then, you and I are
living in the same colony.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
? x, y, z ? A
? R is Transitive.
Hence, R is reflexive, symmetric and transitive.
1 C. Question
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:
R = {(x, y) : x is wife of y}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x is wife of y}
Check for Reflexivity:
Since x and x are the same people.
Then, x cannot be the wife of itself.
A person cannot be a wife of itself.
Wendy is the wife of Sam; Wendy can’t be the wife of herself.
So, ? x ? A, then (x, x) ? R.
? R is not reflexive.
Check for Symmetry:
If x is the wife of y.Then, y cannot be the wife of x.
If Wendy is the wife of Sam, then Sam is the husband of Wendy.
Sam cannot be the wife of Wendy.
So, if (x, y) ? R, then (y, x) ? R.
? x, y ? A
? R is not symmetric.
Check for Transitivity:
If x is the wife of y and y is the wife of z, which is not logically possible.
Then, x is not the wife of z.
It’s easy, take Wendy, Sam, and Mac.
If Wendy is the wife of Sam, Sam can’t be the wife of Mac.
Thus, the possibility of Wendy being the wife of Mac also eliminates.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
? x, y, z ? A
? R is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive.
1 D. Question
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:
R = {(x, y) : x is father of y}
Answer
We have been given that,
A is the set of all human beings in a town at a particular time.
Here, R is the binary relation on set A.
So, recall that
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
Using these criteria, we can solve these.
We have,
R = {(x, y): x is father of y}
Check for Reflexivity:
Since x and x are the same people.
Then, x cannot be the father of itself.
A person cannot be a father of itself.
Leo is the father of Thiago
So, ? x ? A, then (x, x) ? R.
? R is not reflexive.
Check for Symmetry:
If x is the father of y.
Then, y cannot be the father of x.
If Sam is the father of Mac, then Mac is the son of Sam.
Mac cannot be the father of Sam.
So, if (x, y) ? R, then (y, x) ? R.
? x, y ? A
? R is not symmetric.
Check for Transitivity:
If x is the father of y and y is the father of z, then, x is not the father of z.
Take Mickey, Sam, and Mac.
If Mickey is the father of Sam, and Sam is the father of Mac.
Thus, Mickey is not the father of Mac, but the grandfather of Mac.
So, if (x, y) ? R and (y, z) ? R, then (x, z) ? R.
? x, y, z ? A
? R is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive.
2. Question
Relations R
1
, R
2
, R
3
 and R
4
 are defined on a set A = {a, b, c} as follows :
R
1
 = {(a, a) (a, b) (a, c) (b, b) (b, c), (c, a) (c, b) (c, c)}
R
2
 = {(a, a)}
R
3
 = {(b, a)}
R
4
 = {(a, b) (b, c) (c, a)}
Find whether or not each of the relations R
1
, R
2
, R
3,
 R
4
 on A is (i) reflexive (iii) symmetric (iii) transitive.
Answer
We have set,
A = {a, b, c}
Here, R
1
, R
2
, R
3,
 and R
4
 are the binary relations on set A.
So, recall that for any binary relation R on set A. We have,
R is reflexive if for all x ? A, xRx.
R is symmetric if for all x, y ? A, if xRy, then yRx.
R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.
So, using these results let us start determining given relations.
We have
R
1
 = {(a, a) (a, b) (a, c) (b, b) (b, c) (c, a) (c, b) (c, c)}
(i). Reflexive:
For all a, b, c ? A. [? A = {a, b, c}]
Then, (a, a) ? R
1
(b, b) ? A
(c, c) ? A
[? R
1
 = {(a, a) (a, b) (a, c) (b, b) (b, c) (c, a) (c, b) (c, c)}]
So, ? a, b, c ? A, then (a, a), (b, b), (c, c) ? R.
? R
1
 is reflexive.
(ii). Symmetric:
If (a, a), (b, b), (c, c), (a, c), (b, c) ? R
1
Then, clearly (a, a), (b, b), (c, c), (c, a), (c, b) ? R
1
? a, b, c ? A
[? R
1
 = {(a, a) (a, b) (a, c) (b, b) (b, c) (c, a) (c, b) (c, c) }]
But, we need to try to show a contradiction to be able to determine the symmetry.
So, we know (a, b) ? R
1