Mathematics Previous Year NDA Questions Of Relation

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Previous Year NDA Questions Of Relation

Q 2783580447

Let `S` be the set of all persons living in Delhi. We say that x, y in S are related if they were born m Delhi on the same day. Which one of the following is correct ?
NDA Paper 1 2017

(A)

The relation is an equivalent relation

(B)

The relation is not reflexive but it is symmetric and transitive

(C)

The relation is not synunetric but it is reflexive and transitive

(D)

The relation is not- transitive but it is reflexive and symmetric

Solution:

`S= xRy ` (R if `x` and `y` were born in Delhi)

Reflexive Relation

(i) `=> xRx ` (`x` was born in Delhi)

(So relation is reflexive)

(ii) `=> yRx` (`y` & `x` also born in Delhi , if `x` & `y` born in Delhi)

(`R` is symmetric)

(iii) `=> xRy`and `yR_2` So `xR_2`

(If x & y were born in Delhi and y & z were born in Delhi then x & z also born in Delhi)

(Here R is Transitive)

(iv) `=> ` So R will be equivalance Relation.

Correct Answer is `=>` (A) The relation is an equivalent relation

Q 2136478372

Suppose, there is :a relation * between the positive
numbers `x` and `y` given by `x` * `y` if and only if `x <= y^( 2)`.
Then which one of the following is correct?
NDA Paper 1 2016

(A)

* is reflexive but not transitive and symmetric

(B)

* is transitive but not reflexive and symmetric

(C)

* is symmetric and reflexive but not transitive

(D)

* is symmetric but not reflexive and transitive

Solution:

(i) Reflexive
Given, `xRy => x` is less than `y^(2)` .

`:. xRx => x` is less than `x^(2)`

Which is not true for `x=1/2`.

Hence, R is reflexive.

(ii) Symmetric
`xRy => x` is less than `y^(2)` .

`yRx => y` is less than `x^(2)` is not true because

`1R2 => 1` is less than `2^(2)`

`2R1 => 2` is less than `1^(2)`

It is not symmetric.

R is transitive

Hence, option (2) is correct.

Correct Answer is `=>` (B) * is transitive but not reflexive and symmetric

Q 2361780625

Let `R` be a relation on the set N of natural numbers defined by `nRm <=>n` is a factor of `m`. Then, which one of the following is correct?
NDA Paper 1 2016

(A)

R is reflexive, symmetric but not transitive

(B)

R is transitive, symmetric but not reflexive

(C)

R is reflexive, transitive but not symmetric

(D)

R is an equivalence relation

Solution:

Given, R is a relation on the set N of natural numbers defined by `nRm <=> n` is a factor of m.

Reflexive Since, n is a factor of n for each `n in N`,
therefore `nRn, AA n in N`, i.e. R is reflexive.

Symmetric Note that 2 is a factor of 4 but 4 is not a factor of 2, i.e. 2R4 but 4R2.

Thus, R is not symmetric.

Correct Answer is `=>` (C) R is reflexive, transitive but not symmetric

Q 1688223107

Let `X` be the set of all persons living in a city.
Persons `x, y` in `X` are said to be related as `x < y`, if
`y` is atleast `5` yr older than `x`. Which one of the
following is correct?
NDA Paper 1 2015

(A)

The relation is an equivalence relation on `X`

(B)

The relation is transitive but neither reflexive nor symmetric

(C)

The relation is reflexive but neither transitive nor symmetric

(D)

The relation is symmetric but neither transitive nor reflexive

Solution:

We have, `X =` Set of all persons living in a city

Let `R` be a relation on `X`, define as `x < y`, if y is atleast

`5` yr older than `x`.

Clearly, `x != x`, so `R` is not reflexive.

Now, let `xRy`, then `x < y` i.e. `y` is atleast `5` yr older than

`x`.

Thus, `x` is smaller than `y`, so `y R x`. Hence, `R` is not

symmetric.

Now, let `xRy` and `yRz`, then `x < y` and `y < z`, clearly `x < z`.

Hence, `R` is transitive.

Correct Answer is `=>` (B) The relation is transitive but neither reflexive nor symmetric

Q 2230134012

Let `X` be the set of all persons living in Delhi. The persons a
and `b` in `X` are said to be related, if the difference in their
ages is atmost `5 yr`. The relation is
NDA Paper 1 2015

(A)

an equivalence relation

(B)

reflexive and transitive but not symmetric

(C)

symmetric and transitive but not reflexive

(D)

reflexive and symmetric but not transitive

Solution:

Given, `R = {(a, b) : |a - b| <= 5}`

For reflexive `(a, a) in R`

`(a, a) = | a - a | = 0 , 0 <= 5, (a, a) in R`

Hence, `R` is reflexive.

For symmetric `(a, b) in R => (b, a) in R`

`(a, b) = | a - b | <= 5, (b, a) = | b - a | <= 5`

`:. (a, b) in R => (b ,a) in R`

Hence, `R` is symmetric.

For transitive `(a, b) in R, (b, c) in R => (a, c) in R`

`(a ,b) = | a - b | <= 5,(b, c)= |b-c| <= 5`

`Ia - b | + | b - c| <= 10, | a - c | <= 10`

` :. (a , c) notin R`

Hence, `R` is not transitive.

So, `R` is reflexive and symmetric but not transitive.

Correct Answer is `=>` (D) reflexive and symmetric but not transitive

Q 1678423306

Let `Z` be the set of integers and `aRb`, where `a, b in Z`
if and only if `(a -b)` is divisible by `5`.

Consider the following statements

I. The relation R partitions Z into five equivalent
classes.

II. Any two equivalent classes are either equal or
disjoint.

Which of the above statements is/are correct?
NDA Paper 1 2015

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

We have, `aRb`, where `a, b in Z`, if `(a- b)` is divisible

by `5`.

Thus, `(a- b)= 0, 5, 10, 15, ....`

So, Statement `I` is true, since the relation `R` partitions `Z`

into five equivalent classes, which are, `0, 1, 2, 3, 4`.

Also, any two equivalent classes are disjoint, but not

equal.

Hence, both the statements are true.

Correct Answer is `=>` (C) Both I and II

Q 1648123903

`Let A = {x,y,z}` and `B= {p,q,r,s}`, what is the

number of distinct relations from `B` to `A?`
NDA Paper 1 2015

(A)

`4096`

(B)

`4094`

(C)

`128`

(D)

`126`

Solution:

Number of distinct relations from

`B` to `A = 2^(mn) = 2^(4 xx 3) = 2^(12) = 4096`

Correct Answer is `=>` (A) `4096`

Q 1762280135

Let `X` be the set of all citizens of India. Elements `x, y` in
`X` are said to be related, if the difference of their age is
`5` yr. Which one of the following is correct?
NDA Paper 1 2014

(A)

The relation is an equivalence relation on `X`

(B)

The relation is symmetric but neither reflexive nor transitive

(C)

The relation is reflexive but neither symmetric nor transitive

(D)

None of the above

Solution:

Given that,

`X =` {Set of all citizens of India}

and `R = { (x, y) : x, y in X, |x - y| = 5}`

(i) Reflexive `|x - x | = 0 != 5`

`:. xRx notin R`

So, `R` is not reflexive.

(ii) Symmetric Again,

`xRy => |x-y| =5 => |y-x| =5`

`=> yRx`

So, Fl is symmetric

(iii) Transitive Let `x, y, z in X,`

then `x R y => | x - y| = 5`

and `y R z =>| y - z| = 5`

But `|x- z| != 5`

So, `R` is not transitive.

Hence, the relation is symmetric but neither reflexive nor

transitive.

Correct Answer is `=>` (B) The relation is symmetric but neither reflexive nor transitive

Q 1772280136

Consider the following relations from `A` to `B`, where
`A = {u, v, w, x, y, z}` and `B = {p, q, r, s}`.

I. `{(u, p), (v, p), (w, p), (x, q), (y, q), (z, q)}`
II. `{(u, p), (v, q), (w, r), (z, s)}`
III. `{(u, s), (v, r), (w, q), (u, p), (v, q), (z, q)}`
IV. `{(u, q), (v, p), (w, s), (x, r), (y, q), (z, s)}`

Which of the above relations are not functions?
NDA Paper 1 2014

(A)

`1` and `2`

(B)

`1` and `4`

(C)

`2` and `3`

(D)

`3` and `4`

Solution:

Given that, `A= {u, v, w, x, y, z}`

and `B = {p, q, r, s}`

We know that, a mapping `f : x -> y` is said to be a function,

if each element in the set `x` has its image in set `y`. It is also

possible that there are few elements in set `y` which are not

the image of any element in set `x`. Every element in set `x`

should have one and only one image.

Correct Answer is `=>` (C) `2` and `3`

Q 1702280138

Let `S` denote set of all integers. Define a relation `R` on `S`
as `'aRb` if `ab >= 0`, where `a, b in S'`. Then, `R` is
NDA Paper 1 2014

(A)

reflexive but neither symmetric nor transitive relation

(B)

reflexive, symmetric but not transitive relation

(C)

an equivalence relation

(D)

symmetric but neither reflexive nor transitive relation

Solution:

Given that, `S =` Set of all integers.

and `R = {(a,b) , a b in S` and `ab >= 0}`.

(i) Reflexive `a R a => a . a = a^2 >= 0 AA a in S`

So, `R` is reflexive.

(ii) Symmetric `a R b => ab => 0 AA a, b in S`

`=> b.a >= 0 => b R a`

So, `R` is symmetric.

(iii) Transitive If `a R b => ab >= 0`

and `bRc => bc >= 0AA a,b,c in S`,

then `ac >= 0 => aRc`.

So, `R` is also transitive.

Hence, `R` is an equivalence relation.

Correct Answer is `=>` (C) an equivalence relation

Q 1762334235

The relation `S` is defined on the set of integers `Z` as `xSy`,
if integer `x` divides integer `y`. Then,
NDA Paper 1 2014

(A)

`S ` is an equivalence relation

(B)

`S ` is only reflexive and symmetric

(C)

`S ` is only reflexive and transitive

(D)

`S ` is only symmetric and transitive

Solution:

The relation `S` is defined on the set of integers

`Z` and `xSy` if integer `x` divides integer `y`.

Reflexive Since, every integer divides itself

`:.` Integer `x` divides integer `x`

`=> xSx`

Hence, `S` is reflexive.

Symmetric Let `x, y in z` such that `xSy`

i.e., integer `x` divides integer `y`

Now, this does not implies that integer `y` divides integer `x`.

e.g., Take `x = 2` and `y = 4`

Then, `2` divides `4` but `4` does not divides `2`.

Thus, `S` is not symmetric.

Transitive Let `x, y, z in Z` such that `xSy` and `ySz`.

`=>` Integer `x` divides integer `y` and integer `y` divides integer `z`

`=>` Integer `x` divides integer `z`

`=> xSz`

Hence, `S` is transitive.

Correct Answer is `=>` (C) `S ` is only reflexive and transitive

Q 2342545433

If `A= {1, 2}, B = {2, 3}` and `C = {3, 4}`, then what is
the cardinality of `(A xx B) cap (A xx C)?`
NDA Paper 1 2013

(A)

`8`

(B)

`6`

(C)

`2`

(D)

`1`

Solution:

Given that, `A = {1,2},8 = {2, 3}` and `C = {3, 4}`

Now, `(A xx B) = { 1, 2} xx {2, 3}`

`= {(1, 2), (1, 3), (2, 2), (2, 3)}`

and `(A xx C) = {1,2} xx {3,4}`

`={(1,3),(1,4),(2,3),(2,4)}`

`:. (A xx 8) cap (A xx C) = {(1, 3), (2, 3)}`

So, the cardinality of `(A xx B) cap (A xx C)` is `2`.

Correct Answer is `=>` (C) `2`

Q 2352545434

If `A` is a finite set having `n` elements, then the
number of relations which can be defined in `A` is
NDA Paper 1 2013

(A)

`2^n`

(B)

`n^2`

(C)

`2^(n^2)`

(D)

`n^n`

Solution:

If `A` is a finite set having `n` elements, then the number of

relations which can be defined in `A` is `2^(n xx n) = 2^(n^2)`.

Correct Answer is `=>` (C) `2^(n^2)`

Q 2372545436

The relation `R` in the set `Z` of integers given by
`R ={(a, b) : a- b` is divisible by `5`} is
NDA Paper 1 2013

(A)

reflexive

(B)

reflexive but not symmetric

(C)

symmetric and transitive

(D)

an equivalence relation

Solution:

Given, `R ={(a, b) : a - b` is divisible by `5`}

Reflexive `(a - a)` is divisible by `5` for all `a in z`. So, `R` is reflexive.

Symmetric Let `(a, b) in R => (a - b)` is divisible by `5`.

`=> (b - a)` is divisible by `5`.

`=> b - a in R`

So, `R` is symmetric.

Transitive Let `(a, b) in R` and `(b, c) in R`

`=> (a - b)` and `(b - c)` are both divisible by `5`.

`=> a - b + b - c` is divisible by `5`.

`=> (a - c)` is divisible by `5`.

`=> (a, c) in R`

So, `R` is transitive.

Thus, `R` is reflexive, symmetric and transitive.

Hence, `R` is an equivalence relation.

Correct Answer is `=>` (D) an equivalence relation

Q 2312545439

If `A = {a, b, c, d}` and `B = {x, y, z}`, then what is the
number of elements in `A xx B` ?
NDA Paper 1 2013

(A)

`6`

(B)

`7`

(C)

`12`

(D)

`64`

Solution:

Given that, `A = {a, b, c, d}` and `B = {x, y, z}`

Here, `n (A) = 4` and `n (B) = 3`

`:. n (A xx B ) = n (A) xx n (B)`

`= 4 xx 3 = 12`

Correct Answer is `=>` (C) `12`

Q 2355112964

If `A={x,y},B={2,3}` and `C={3,4}`, then what is
the number of elements in `A xx (B cup C)`?
NDA Paper 1 2013

(A)

`2`

(B)

`4`

(C)

`6`

(D)

`8`

Solution:

Given that, `A ={x,y}, B = {2, 3}`

and `C = {3, 4}`

` B cup C= (2,3) cup (3,4) = (2,3,4)`

and `A xx (B cup C) = (x,y) xx (2,3,4)`

`= { (x,2),(x, 3),(x, 4),(y,2),(y, 3),(y, 4)}`

`:.` Number of elements in `A xx (B cup C)`

ie, `n{A xx (B cup C)} = 6`

Correct Answer is `=>` (C) `6`

Q 2385423367

If `A= {0, 1}` and `B = {1, 0}`, then what is the value
of `A xx B`?
NDA Paper 1 2013

(A)

`{ (0, 1), (1, 0)}`

(B)

`{(0, 0), (1, 1)}`

(C)

`{(0, 1), (1, 0), (1, 1)}`

(D)

`A xx A`

Solution:

Given that, `A= {0, 1}` and `a = {1, 0}`

Then, `A xx B = {0,1} xx {1,0}`

and `= { (0, 1), (0, 0), (1, 1) (1, 0)}`

`A xx A = {0, 1} xx { 0, 1}`

`= {(0, 0), (0, 1), (1, 0), (1, 1)}`

`:. A xx B = A xx A`

Correct Answer is `=>` (D) `A xx A`

Q 2355534464

If `A ={1, 2, 5, 6}` and `B ={I, 2, 3}`, then what is
`(A xx B) cap (B xx A)` equal to?
NDA Paper 1 2011

(A)

`{(1, 1), (2, 1), (6, 1), (3, 2)}`

(B)

`{(1, 1), (1, 2), (2, 1), (2, 2)}`

(C)

`{(1, 1), (2, 2)}`

(D)

`{(1, 1), (1. 2), (2, 5) (2, 6)}`

Solution:

`∵ A = {1, 2, 5, 6}` and `B = {1, 2, 3}`

`:. A xx B = {(1, 1), (1, 2), (1, 3), (2, 1), (2,2), (2, 3)`,

`(5, 1), (5, 2), (5, 3), (6, 1), (6, 2), (6, 3)}`

`B xx A = { (1, 1), (1, 2), (1, 5), (1, 6), (2, 1), (2, 2),`

`(2, 5), (2, 6), (3, 1), (3, 2), (3, 5), (3, 6)}`

`=> (A xx B) cap (B xx A) = { (1, 1), (1, 2), (2, 1), (2, 2)}`

Correct Answer is `=>` (B) `{(1, 1), (1, 2), (2, 1), (2, 2)}`

Q 2305345268

Consider the following with regard to a relation `R`
on a set of real numbers defined by `x Ry` if and
only if `3x + 4y = 5`

I. 0 RI II. IR `1/2`
III. `2/3` R `3/4`

Which of the above are correct?
NDA Paper 1 2011

(A)

I and II

(B)

I and III

(C)

II and III

(D)

I, II and III

Solution:

Since, on the set of real numbers, R is a relation

defined by `x Ry` if and only if `3x + 4 y = 5` for which `1R 1/2` and

`2/3` R `3/4`.

i.e., `1R 1/2 => 3·1 + 4·1/2 = 5`,

and `2/3 R 3/4 => 3 xx 2/3 + 3/4 xx 4 = 5`

Hence, both the Statements II and III are correct.

Correct Answer is `=>` (C) II and III

Q 2375445366

Let M be the set of men and R is a relation `text('is son of')` defined on M. Then, R is
NDA Paper 1 2011

(A)

an equivalence relation

(B)

a symmetric relation only

(C)

a transitive relation only

(D)

None of the above

Solution:

`M` = Set of men and R is a relation 'is son of' defined

on `M`.

Reflexive relation `aRa`.

Since, a cannot be a son of a.

Symmetric relation

`aRb => bRa`

which is also not possible.

Transitive relation `aRb, bRc => eRa`

which is not possible.

Correct Answer is `=>` (D) None of the above

Q 2305845768

The relation `R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3),`
`(1, 3)}` on a set `A= {1, 2, 3}` is
NDA Paper 1 2010

(A)

reflexive, transitive but not symmetric

(B)

reflexive, symmetric but not transitive

(C)

symmetric, transitive but not reflexive

(D)

reflexive but neither symmetric nor transitive

Solution:

`∵ R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}`

Reflexive

`∵ 1R1, 2R2, 3R3`

Hence, `R` is a reflexive relation.

Symmetric

`∵ 1R2` but `2R1`

Hence, `R` is not a symmetric relation.

Transitive

`∵ 1R2,2R3 => 1R3`

Hence, `R` is a transitive relation.

Correct Answer is `=>` (A) reflexive, transitive but not symmetric

Q 2305356268

If `A= {2, 3}, B = {4, 5}` and `C = {5, 6}`, then what is
the number of elements in `A xx (B cap C)`?
NDA Paper 1 2010

(A)

`2`

(B)

`4`

(C)

`6`

(D)

`8`

Solution:

`∵ A = {2, 3}, B = { 4, 5}` and `C = { 5, 6}`

`:. B cap C = {5}`

`=> A xx {B cap C} = {(2, 5), (3, 5)}`

Hence, the required number of elements in `A xx (B cap C) = 2`.

Correct Answer is `=>` (A) `2`

Q 2375056866

The order of a set `A` is `3` and that of a set `B` is `2`.
What is the number of relations from `A` to `B`?
NDA Paper 1 2010

(A)

`4`

(B)

`6`

(C)

`32`

(D)

`64`

Solution:

Here, `n(A) = 3` and `n(B) = 2`

`:.` Number of relations from `A` to `B = 2^([n(A) xx n(B)])`

`= 2^((3 xx 2)) = 2^6`

`= 64`

Correct Answer is `=>` (D) `64`

Q 2355567464

Let `X` be the set of all graduates in India. Elements
`x` and `y` in `X` are said to be related, if they are
graduates of the same university.
Which one of the following statements is correct?
NDA Paper 1 2010

(A)

Relation is symmetric and transitive only

(B)

Relation is reflexive and transitive only

(C)

Relation is reflexive and symmetric only

(D)

Relation is reflexive symmetric and transitive

Solution:

`xRy <=> x` and `y` are graduates of the same

university.

Reflexive `xRx <=> x` and `y` are graduates of the same university.

So, relation is reflexive.

Symmetric `xRy <=> x` and `y` are graduates of the same

university.

`=> yRx <=> y` and `x` are graduates of the same university.

Hence, relation is symmetric.

Transitive `xRy, yRz <=> xRz`

It means `x` and `y, y` and `z` are graduates of the same university,

then `x` and `z` are also graduates of the same university.

So, relation is transitive.

Hence, relation is reflexive, symmetric and transitive.

Correct Answer is `=>` (D) Relation is reflexive symmetric and transitive

Q 2335667562

If `A= {a, b, c, d}`, then what is the number of
proper subsets of `A?`
NDA Paper 1 2010

(A)

`16`

(B)

`15`

(C)

`14`

(D)

`12`

Solution:

`∵ n(A) = 4`

`:.` Number of subsets `= 2^n = 2^4`

Hence, number of proper subsets `= 2^4 - 1`

`= 16 - 1 = 15`

Correct Answer is `=>` (B) `15`

Q 2325678561

Let `A= { -1, 2, 5, 8}, B = {0, 1, 3, 6, 7}` and `R` be the
relation `text('is one less than')` from `A` to `B`, then how
many elements will `R` contain?
NDA Paper 1 2009

(A)

`2`

(B)

`3`

(C)

`5`

(D)

`9`

Solution:

Given, `A= { -1, 2, 5, 8}` and `B = {0, 1, 3, 6, 7}`

`:. R = {(-1, 0), (2, 3), (5, 6)}`

(`∵ R = A` is one less than from `B`)

Hence, total number of elements in `R` is `3`.

Correct Answer is `=>` (B) `3`

Q 2345480363

Sets A and B have n elements in common. How
many elements will `(A xx B)` and `(B xx A)` have in
common?
NDA Paper 1 2009

(A)

`0`

(B)

`1`

(C)

`n`

(D)

`n^2`

Solution:

Let `A = {1,2,3}` and `B = {2,3,4}`

`AxxB = {(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,2),(3,3)(3,4)}`

`AxxB = {(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(4,1),(4,2),(4,3)}`

No. of common element ` = {(2,2),(2,3),(3,3),(3,2)} = 2^2`

The total number of elements common in `(A xx B)` and

`(B xx A)` is `n^2` . (by property)

Correct Answer is `=>` (D) `n^2`

Q 2315591460

If `R ={x | x in N, x` is a multiple of `3` and `x <= 100}`
and `S ={x | x in N, x` is a multiple of `5` and `x <= 100}`
What is the number of elements in
`(R xx S) cap (S xx R)`?
NDA Paper 1 2009

(A)

`36`

(B)

`33`

(C)

`20`

(D)

`6`

Solution:

`∵ R = { 3, 6, 9, 12, 15, ... , 99}`

and `S = { 5, 10, 15, ... , 95}`

Now, `(R xx S) cap (S xx R)`

`= (R cap S) xx (S cap R)`

`= (15, 30, 45, 60, 75, 90) xx (15, 30, 45, 60, 75, 90)`

`:.` Number of elements in `(R xx S) cap (S xx R) = 6 xx 6 = 36`

Correct Answer is `=>` (A) `36`

Q 2335691562

If `A={a,b,c}` and `R={(a,a),(a,b),(b,c),(b,b)`,
`(c, c), (c, a)}` is a binary relation on `A`, then which
one of the following is correct?
NDA Paper 1 2009

(A)

R is reflexive and symmetric, but not transitive

(B)

R is reflexive and transitive, but not symmetric

(C)

R is reflexive, but neither symmetric not transitive

(D)

R is reflexive, symmetric and transitive

Solution:

` ∵ (a, a), (b, b), (c, c) in R`

So, R is a reflexive relation.

But `(a, b) in R` and `(b, a), notin R`.

Thus, `R` is not a symmetric relation.

Also,`(a,b),(b,c) in R => (a,c), notin R`

Hence, R is not a transitive relation.

Correct Answer is `=>` (C) R is reflexive, but neither symmetric not transitive

Q 2345891763

If a set A contains 4 elements, then what is the
number of elements in `A xx P(A)`?
NDA Paper 1 2008

(A)

`16`

(B)

`32`

(C)

`64`

(D)

`128`

Solution:

Since, the number of elements in set `A` is `4`.

`:.` Number of elements in `P(A) = 2^4 = 16`

So, the number of elements in `A xx P(A) = 4 xx 16 = 64`.

Correct Answer is `=>` (C) `64`

Q 2356223174

If `A = {1, 2, 3, 4}` and `R = { (1, 1), (1, 3), (2, 2), (3, 1),`
`(3,4),(4,3),(4,4)}` is a relation on `A xx A`, then
which one of the following is correct?
NDA Paper 1 2008

(A)

R is reflexive

(B)

R is symmetric and transitive

(C)

R is transitive, but not reflexive

(D)

R is neither reflexive nor transitive

Solution:

Since, `3 in A`

But `(3, 3) notin R`

So, it is not reflexive.

and `(3, 4) in R` and `(4, 3) in R`

But `(3, 3) notin R`

So, it is also not transitive.

Hence, `R` is neither reflexive nor transitive.

Correct Answer is `=>` (D) R is neither reflexive nor transitive

Q 2316223179

NDA Paper 1 2008

Assertion : `{x in R | x^2 < 0}` is not a set. Here, `R` is the set of real numbers.

Reason : For every real number `x, x^2 >= 0` .

(A)

Both A and R individually true and R is the correct explanation of A

(B)

Both A and R are individually true but R is not the correct explanation of A

(C)

A is true but R is false

(D)

A is false but R is true

Solution:

Both (A) and (R) are true and (R) is the correct

explanation of (A).

`x^2` is never negative but it can be positive or zero.

Correct Answer is `=>` (A)

Q 2316323270

Let `R` be the relation defined on the set of natural
number `N` as `aRb ; a, b in N`, if `a` divides `b`. Then,
which one of the following is correct?
NDA Paper 1 2008

(A)

R is reflexive only

(B)

R is symmetric only

(C)

R is transitive only

(D)

R is reflexive and transitive

Solution:

For reflexive

`aRa => a` divides a

So, `R` is reflexive.

For symmetric

`aRb => a` divides `b`

`bRa => b` divides `a`

which may not be possible.

Hence, `R` is not symmetric.

For transitive

`aRb => a` divides `b => b = ka`

`bRc => b` divides `c => c = lb`

Now, `c = lka`

`=> a` divides `c => aRc => aRb, bRc => cRa`

Thus, `R` is transitive.

Correct Answer is `=>` (D) R is reflexive and transitive

Q 2316323279

Let `N` be the set of integers. `A` relation `R` on `N` is
defined as `R = {(x, y) | xy > 0, x, y in N}`. Then,
which one of the following is correct?
NDA Paper 1 2007

(A)

R is symmetric, but not reflexive

(B)

R is reflexive, but not symmetric

(C)

R is symmetric and reflexive, but not transitive

(D)

R is an equivalence relation

Solution:

`∵ R = {(x, y) | xy > 0, x, y in N}`

Reflexive

`∵ x, y in N`

`:. x, x in N => x^2 > 0`

So, `R` is reflexive.

Symmetric

`∵ x ,y in N` and `xy > 0 => yx > 0`

Hence, `R` is also symmetric.

Transitive

`∵ x, y, z in N => xy > 0 , yz > 0 => xz > 0`

So, `R` is also transitive.

Thus, `R` is an equivalence relation.

Correct Answer is `=>` (D) R is an equivalence relation

Q 2386723677

A relation R is defined over the set of
non-negative integers as `xRy => x^2 + y^2 = 36`.
What is `R`?
NDA Paper 1 2007

(A)

`{(0, 6)}`

(B)

`{(0, 6), (sqrt(11), 5), (3, 3sqrt(3))}`

(C)

`{(6, 0), (0, 6)}`

(D)

`{(sqrt(11), 5), (2, 4sqrt(2)), (5, sqrt(11)), (4 sqrt(2), 2)}`

Solution:

Since, `R` is defined over the set of non-negative

integers, then `R = {(6, 0), (0, 6)}` because the options (b) and (c)

does not have integers in all pairs. Also in options (a) and (c), the

most appropriate option is (c).

Correct Answer is `=>` (C) `{(6, 0), (0, 6)}`

Q 2356823774

Consider the following statements
I. Parallelism of lines is an equivalence relation
II. `xRy`, if `x` is a father of `y`, is an equivalence relation
Which of the statements given above is/are correct?
NDA Paper 1 2007

(A)

Only I

(B)

Only II

(C)

Both I and II

(D)

Neither I nor II

Solution:

I. Let `l, m, n` are parallel lines and `R` is a relation.

`:. l || l`. then `R` is reflexive.

and `l || m` and `m || l`, the `R` is symmetric.

Also, `l || m , m || n => l || n`,

then `R` is transitive.

Hence, `R` is an equivalence relation.

II. If `x` is father of `y` and `y` is not father of `x,` then relation is not

symmetric, thus relation is not equivalence.

Correct Answer is `=>` (A) Only I

Q 2386823777

Let `A = { x in W`, the set of whole numbers and
`x < 3}, B = {x in N`, the set of natural numbers and
`2 <= x < 4}` and `C = {3, 4}`, then how many elements
will `(A cup B) xx C` contain?
NDA Paper 1 2007

(A)

`6`

(B)

`8`

(C)

`10`

(D)

`12`

Solution:

Given, `A= {x in W`, the set of whole numbers and

`X < 3} = {0, 1, 2}`

`B = {x in N`, the set of natural numbers and `2 <= x < 4}`

`= { 2, 3}`

`C = {3, 4}`

`A cup B = {0, 1, 2, 3}`

`(A cup B) xx C = { 0, 1, 2, 3} xx {3, 4}`

`= {(0, 3), (0, 4), (1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}`

So, required number of elements containing by `(A cup B) xx C` is `8`.

Correct Answer is `=>` (B) `8`