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Tricks & Tips of Sets for NDA

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Cardinality of set

Cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3.

Note : Same element in set will be count only once.
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 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`

Conversion from roster form to set builder and vice versa

See examples
Q 1954823754

Write the following sets in the set-builder form:

(i) `{3, 6, 9, 12}`

(ii) `{2, 4, 8, 16, 32}`

(iii) `{5, 25, 125, 625}`

(iv) `{2, 4, 6, .....}`

(v) `{1, 4, 9, ... 100}`

Class 11 Exercise 1.1 Q.No. 4
Solution:

(i) `A= { x : x = 3n , n in N` and `1 le n le 4}`

(ii) `B = { x : x = 2^n , n in N` and `1 le n le 5}`

(iii) `C= { x : x = 5^n , n in N` and `1 le n le 4}`

(iv) `D = { x : x` is an even natural number}

(v) `E = { x : x = n^2 , n in N` and `1 le n le 10}`
Q 1914723650

Write the following sets in roster form :

(i) `A = {x: x` is an integer and `-3 < x < 7}`

(ii) `B = {x: x` is a natural number less than `6}`

(iii) `C = {x: x` is a two-digit natural number such that the sum of its digits is `8}`

(iv) `D = {x : x` is a prime number which is divisor of `60}`

(v) `E =` The set of all letters in the wod `text ('TRIGONOMETRY')`

(vi) `F =` The set of all letters in the word `text ('BETTER')`
Class 11 Exercise 1.1 Q.No. 3
Solution:

(i) `A= {-2 , 1 , 0 , 1 , 2 , 3 , 4, 5 , 6 }`

(ii) `B = { 1 , 2, 3 , 4 ,5 }`

(iii) `C= { 17 , 26 , 35 , 44 , 53 , 62 , 71 ,80}`

(iv) `D= {2 , 3, 5}`

(v) `E = {T, R, I, G, 0, N, M, E, Y}`

(vi) `F= { B, E, T, R}`.
Q 1944023853

List all the elements of the following sets :

(i) `A = { x : x` is an odd natural number}

(ii) `B = { x: x` is an integer, `-1/2 < x < 9/2}`

(iii) `C = {x : x` is an integer, `x^2 le 4}`

(iv) `D = { x : x` is a letter in the word `text ('LOYAL')}`

(v) `E = {x: x` is a month of a year not having `31` days}

(vi) `F = {x : x` is a consonant in the English alphabet which precedes `k}`.

Class 11 Exercise 1.1 Q.No. 5
Solution:

(i) `A= { 1 , 3 , 5, ... }`

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

(iii) `C= {-2,-1,0, 1,2}`

(iv) `D = {L, O, Y,A}`

(v) `E = text ( { February, April, June, September, November} )`

(vi) `F= { b ,c ,d , J , g , h , j }`.
Q 1713678540

Write the following sets in the roaster form.

(i)` D = { t | t^3 =t , t in R}`

(ii)` E = {w | (w-2)/(w+3) = 3, w in R}`

(iii) `F ={ x | x^4 - 5x^2 + 6 = 0, x in R}`
NCERT Exemplar
Solution:

1.We have ` 0 = {t | t^3 = t , t in R}`

` :. t^3 = t`

`=> t^3 -t = 0 => t (t^2 -1) = 0`

` => t (t -1) (t + 1) = 0 => t = 0, 1, - 1`

` :. D = {-1, 0, 1}`

2. We have ` E = { w | (w-2)/(w + 3) = 3 , w in R}`

`:. (w -2) /(w +3) =3`

` => w -2 = 3w + 9 => w - 3w = 9 + 2 `

` => -2w = 11 => w = (-11)/2 `

`:. E = {(-11)/2 }`

3. We have ` F ={x | x^ 4 - 5x^2 + 6 = 0, x in R}`

` :. x^ 4- 5x^2 + 6 = 0`

` => x^ 4 - 3x^2 - 2 x^ 2 + 6 = 0`

` => x^ 2 (x^ 2 - 3)- 2 (x^ 2 - 3) = 0`

` => (x^2 - 3) (x^2 - 2) = 0`

`=> x = pm sqrt(3) , pm sqrt(2)`

` F = {- sqrt(3) , - sqrt(2) , sqrt(2) , sqrt(3) }`

Determining type of set

`"Empty Set or Null Set:"`

A set which does not contain any element is called an empty set, or the null set or the void set and it is denoted by ∅ and is read as phi. In roster form, ∅ is denoted by {}. An empty set is a finite set, since the number of elements in an empty set is finite, i.e., 0.

`"Singleton Set:"`
A set which contains only one element is called a singleton set.
For example: A = {x : x is neither prime nor composite}

`"Finite Set"`:

A set which contains a definite number of elements is called a finite set. Empty set is also called a finite set.
For example: The set of all colors in the rainbow.

`"I n f i n i te Set ::`

The set whose elements cannot be listed, i.e., set containing never-ending elements is called an infinite set.
For example: Set of all points in a plane.

`"Equal sets:"`

Two sets A and B are said to be equal if they contain the same elements. Every element of A is an element of B and every element of B is an element of A.

For example:
A = {p, q, r, s} , B = {p, s, r, q}
Therefore, A = B

`"Power set :"`

The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.
E.g. If `A = { 1, 2 }`, then `P( A ) = { φ,{ 1 }, { 2 }, { 1,2 }}` Also, note that `n [ P (A) ] = 4 = 2^2`

In general, if `A` is a set with `n(A) = m`, then it can be shown that `n [ P(A)] = 2^m`
Q 1974245156

From the sets given below, select equal sets:

`A= {2, 4, 8, 12}`,

`B = {1, 2 , 3 , 4 }`,

`C= {4, 8, 12, 14}`,

`D = {3, 1, 4, 2}`,

`E = {-1, 1 }, F = {0, a}` ,

`G = { 1 , - 1 } , H = { 0 , 1 }`


Class 11 Exercise 1.2 Q.No. 6
Solution:

(i) `B = D `,

(ii) `E = G`
Q 2362545435

Which one of the following is an example of
non-empty set?
NDA Paper 1 2013
(A)

Set of all even prime numbers

(B)

(`x : x^2 - 2 = 0` and `x` is rational)

(C)

{`x : x` is a natural number, `x < 8` and simultaneously `x > 12`}

(D)

{`x : x` is a point common to any two parallel lines}

Solution:

Let `S` be the set of all even prime numbers.

`:. S = {2} =` Non-empty set
Correct Answer is `=>` (A) Set of all even prime numbers
Q 1984034857

State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the `x`-axis.

(ii) The set of letters in the English alphabet.

(iii) The set of numbers which are multiple of `5`.

(iv) The set of animals living on the earth.

(v) The set of circles passing through the origin `(0, 0)`.


Class 11 Exercise 1.2 Q.No. 3
Solution:

(i) Infinite

(ii) Finite

(iii) Infinite

(iv) Finite

(v) Infinite
Q 2315280160

If `P(A)` denotes the power set of `A` and `A` is the
void set, then what is number of elements in
`P{P{P{P(A)} }}`?
NDA Paper 1 2009
(A)

`0`

(B)

`1`

(C)

`4`

(D)

`16`

Solution:

The number of elements in power set of `A` is `1` .

`:. P{P(A)} = 2^1 = 2`

`=> P{P{P(A)}} = 2^2 = 4`

`=> P{P{P{P(A)}}} = 2^4 = 16`
Correct Answer is `=>` (D) `16`
Q 1914034850

Which of the following sets arc finite or infinite:

(i) The set of months of `2` year

(ii) `{1, 2, 3, ... }`

(iii) `{1, 2, 3, ... , 99, 100}`

(iv) The set of positive integers greater than `100`.

(v) The set of prime numbers less than `99`.
Class 11 Exercise 1.2 Q.No. 2
Solution:

(i) Finite

(ii) Infinite

(iii) Finite

(iv) Infinite

(v) Finite.
Q 2833445342

Total numher of elements in the power set of `A` containing `15` elements is

(A)

`2^(15)`

(B)

`(15)^2`

(C)

`2^(15 - 1)`

(D)

`2^(15) - 1`

Solution:

If a set `A` has `n` elements, then its

power set will contain `2"` elements.

`:.` Total number of elements in is power

set of `A = 2^(15)`
Correct Answer is `=>` (A) `2^(15)`

Subset of set

See examples
Q 2713580449

Let `A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}`. Then the number of subsets of A containing two or three elements is
NDA Paper 1 2017
(A)

`45`

(B)

`120`

(C)

`165`

(D)

`330`

Solution:

`n(A) =10`

Required subset of `A=text()^10 C_2 + text()^10 C_3`

`= 45 + 120 =165`
Correct Answer is `=>` (C) `165`
Q 2616034879

List all the subsets of the set `{ –1, 0, 1 }`.

Solution:

Let `A = { –1, 0, 1 }`. The subset of A having no element is the empty set `φ`. The subsets of A having one element are `{ –1 }, { 0 }, { 1 }`. The subsets of A having two elements are `{–1, 0}, {–1, 1} ,{0, 1}`. The subset of A having three elements of A is A itself. So, all the subsets of A are `φ, {–1}, {0}, {1}, {–1, 0}, {–1, 1}, {0, 1}` and `{–1, 0, 1}`.
Q 2366423375

For non-empty subsets `A, B` and `C` of a set `X` such
that `A cup B = B cap C`, which one of the following is
the strongest inference that can be derived?
NDA Paper 1 2007
(A)

`A = B = C`

(B)

`A subseteq B = C`

(C)

`A = B subseteq C`

(D)

`A subseteq B subseteq C`

Solution:

`∵ A cup B = B cap c`

From above strong inference is

`A subseteq B subseteq C`.

e.g., `A = {a}, B ={a, b},C ={a, b, c}`

`A cup B ={a, b}, B cap C ={a, b}`
Correct Answer is `=>` (D) `A subseteq B subseteq C`

Venn diagram question

See examples
Q 1925534461

Draw appropriate Venn diagram for each of the following :

(i) `(A cup B)'`

(ii) `A' cap B'`

(iii) `(A cap B)'`

(iv) `A' cup B'`
Class 11 Exercise 1.5 Q.No. 5
Solution:

(i) `(A cup B)'`

(ii) `A' cap B'`

(iii) `(A cap B)'`

(iv) `A' cup B'`


All shaded region formed by all horizontal and vertical lines is `A' cup B'`.
Q 2306323278

What does the shaded region in the Venn diagram
given following represent?
NDA Paper 1 2007
(A)

`C cap (A' cap B')`

(B)

`C cup (C' cap A cap B)`

(C)

`C cup (C cap A) cup (C cap B)`

(D)

`C cup (A\ B)`

Solution:

The shaded region in the Venn diagram is

represented by `C cup (C cap A) cup (C cap B)`.
Correct Answer is `=>` (C) `C cup (C cap A) cup (C cap B)`
Q 2773191046

Consider the following in respect of sets A and B :
NDA Paper 1 2017
(A)

`(A - B) cup B = A`

(B)

`(A - B) cup A = A`

(C)

`(A - B) cap B = phi`

(D)

`A subseteq B => A cup B = B`

Solution:

Hence options 2,3,4 are correct.
Correct Answer is `=>` (B) `(A - B) cup A = A`

Operation on set

1. `A-B = A∩B'`
2. `A-(B∩C) = (A-B) ∪ (A-C)`
3. `A-(B∪C) = (A-B) ∩ (A-C)`
4.`A∩(B-C) = (A∩B) - (A∩C)`
Q 1965823765

Let `U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A= {1, 2, 3, 4}`,

`B= {2,4,6,8}` and `C= {3, 4, 5, 6}`. Find,

(i) `A'`,

(ii) `B' `

(iii) `(A cup C)'`

(iv) `(A cup B)' `

(v) `(A')'`

(vi) `(B-C)'`
Class 11 Exercise 1.5 Q.No. 1
Solution:

Here, `U= { 1, 2, 3, 4. 5, 6, 7, 8, 9}`

`A={ 1,2,3,4}`

`B= {2,4,6,8}`

and `C= {3,4,5,6}`

(i) `A'=U- A={5,6,7,8,9}`

(ii) `B' = U- B = {1, 3, 5, 7, 9}`

(iii) `A cup C= {1,2,3,4} cup {3,4,5,6}`

`= {1,2,3,4,5,6}`

Hence `(A cup C )' = U -( A cup C)'= {7, 8, 9}`

(iv) `A cup B = { 1.2,3,4} cup {2,4,6,8}`

`= {1,2, 3,4, 6, 8}`

Hence `(A cup B)'= U-(A cup B)= {5, 7, 9}`

(v) `A' = U-A={5.6,7,8,9}`

Hence `(A')' = U - A'= { 1, 2. 3, 4}`


(vi) `B-C= {2,4,6,8}-{3,4,5,6}={2,8}`

Hence `(B -C)' = U- (B- C)`

`= {1, 3, 4, 5, 6, 7, 9}`
Q 1734312252

If `X` and `Y` are two sets and `X'` denotes the complement of `X`, then
`X cap (X cup Y)'` is equal to
NCERT Exemplar
(A)

`X`

(B)

`Y`

(C)

`phi`

(D)

`X cap Y`

Solution:

`X cap (X cup Y)' = X cap (X' cap Y') quad [∵(A cup B)' = A' cap B']`

` = (X cap X') cap (X cap Y')`

` = phi cap (X cap Y') = phi quad [∵ phi cap A = phi]`
Correct Answer is `=>` (C) `phi`
Q 1714601559

For all sets `A` and `B, (A cup B) - B = A - B.`
NCERT Exemplar
Solution:

`LHS = (A cup B) - B = (A cup B) cap B' quad [∵ A - B =A cap B']`

` = (A cap B') cup (B cap B') = (A cap B') cup phi quad [∵ B cap B' = phi ]`

` = A cap B' quad [ ∵ A cup phi = A]`

` = A - B = RHS`
Q 1714212159

If `A = {1, 3, 5, 7, 9, 11, 13, 15, 17}, B = {2, 4, ... , 18}` and `N` the set of
natural numbers is the universal set, then `(A' cup (A cup B) cup B')` is
NCERT Exemplar
(A)

`phi`

(B)

`N`

(C)

`A`

(D)

`B`

Solution:

` A' cup [(A cup B) cap B] quad [ ∵ A cap (B cup C)= (A cap B) cup (A cap C)]`

` = A' u [(A cap B') cup (B cap B')]`

` =A' u [(A cap B') cup phi ] = A' cup (A cap B')`

` = (A' cup A) cap (A' cup B')`

` = A' cup B' = N cap (A' cup B')`

` =A' cup B'= (A cap B)' quad [ ∵ A cap B = phi ]`

` = phi = N`
Correct Answer is `=>` (B) `N`
 
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