Subset

Q.

How many elements has $P\left(A\right),$ if $A=\varphi ?$

Q. Let A = { a , b }, B = { a , b , c }. Is A ⊂ B? What is A ∪ B?

Q. Examine whether the following statements are true or false: (i) { a , b } ⊄ { b , c , a } (ii) { a , e } ⊂ { x : x is a vowel in the English alphabet} (iii) {1, 2, 3} ⊂{1, 3, 5} (iv) { a } ⊂ { a . b , c } (v) { a } ∈ ( a , b , c ) (vi) { x : x is an even natural number less than 6} ⊂ { x : x is a natural number which divides 36}

Q.

Consider the two statements P: He is intelligent and Q: He is strong. Then the symbolic form of the statement "It is not true that he is intelligent or strong" is

1. $\sim P\vee Q$

2. $\sim P\vee \sim Q$

3. $\sim (P\vee Q)$

4. $\sim P\wedge Q$

Q.

Write each of the following subsets of R as an interval:
$\left(i\right)A=\{x:xϵR,-3<x\le 5\}$

$\left(ii\right)B=\{x:xϵR,-5<x\le -1\}$

$\left(iii\right)C=\{x:xϵR,-2\le x<0\}$

$\left(iv\right)D=\{x:xϵR,-1\le x\le 4\}$

Find the length of each of the above intervals

Q.

Let P(A)=P(B), prove that A=B

Q. Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that (i) A × (B ∩ C) = (A × B) ∩ (A × C) (ii) A × C is a subset of B × D

Q. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces: (i) {2, 3, 4} … {1, 2, 3, 4, 5} (ii) { a , b , c } … { b , c , d } (iii) { x : x is a student of Class XI of your school} … { x : x student of your school} (iv) { x : x is a circle in the plane} … { x : x is a circle in the same plane with radius 1 unit} (v) { x : x is a triangle in a plane}…{ x : x is a rectangle in the plane} (vi) { x : x is an equilateral triangle in a plane}… { x : x is a triangle in the same plane} (vii) { x : x is an even natural number} … { x : x is an integer}

Q. The number of subsets of a set containing n elements is
(a) n
(b) 2ⁿ − 1
(c) n²
(d) 2ⁿ

Q. If the solution set of $|x-k|<2$ is a subset of the solution set of the inequality $\frac{2x-1}{x+2}<1,$ then the number of possible integral value(s) of $k$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q.

Let A be a non-empty set such that A $\times$ A has 9 elements among which are found (-1, 0) and (0, 1), then

1. 

2. 

3. 

4. 

Q. If ${}^{n}P_4$ = 360, find the value of n.

Q. Write down all the subsets of the following sets: (i) { a } (ii) { a , b } (iii) {1, 2, 3} (iv) Φ

Q. If P, Q and R are subsets of a set A, then $R\times (P^c\cup Q^c{)}^c=$

1. $(R\times P)\cap (R\times Q)$
2. $(R\times Q)\cap (R\times P)$
3. $(R\times P)\cup (R\times Q)$
4. $Noneofthese$

Q. 14. All the letters of the word EAMCOT are arranged in different possible ways. Find the number of arrangements in which no two vowels are adjacent to each other.

Q.

Show that the following statement is true by the method of contrapositive p : "If x is an integer and $x^2$ is even, then x is also even"

Q.

Write down all possible subsets of each of the following sets:
(i) $\left\{a\right\}$,
(ii) $\{0,1\}$
(iii) $\{a,b,c\}$
(iv) $\{1,\{1\left\}\right\}$
(v) $\left\{\varphi \right\}$

Q. If $X=8^n-7n-1,n\in NandY=49(n-1),n\in N$, then -

1. X ⊆ Y
2. X = Y
3. None of these
4. Y ⊆ X

Q. Decide, among the following sets, which sets are subsets of one and another: A = { x : x ∈ R and x satisfy x 2 – 8 x + 12 = 0}, B = {2, 4, 6}, C = {2, 4, 6, 8…}, D = {6}.

Q.

Let $A=\{1,\left\{2\right\},\{3,4\},5\}$. Which of the following are incorrect statements?
Rectify each:

$\left(i\right)2ϵA$

$\left(ii\right)\left\{2\right\}\subset A$

$\left(iii\right)\{1,2\}\subset A$

$\left(iv\right)\{3,4\}\subset A$

$\left(v\right)\{1,5\}\subset A$

$\left(vi\right)\left\{\varphi \right\}\subset A$

$\left(vii\right)1\subset A$

$\left(viii\right)\{1,2,3,4\}\subset A$

Q. Find the equation of the circle passing through the points $(4,1)$ and $(6,5)$ and whose centre is on the line $4x+y=16$

Q.

if X = {\(8^n-7n-1/n \in N\)} and Y = {\(49(n-1)/n \in N\)}, then

Q. $A_1,A_2,\cdots ,A_{30}$ are $30$ sets, each having $5$ elements and $B_1,B_2,\cdots ,B_n$ are $n$ sets each with $3$ elements. If $\bigcup _{i=1}^{30}{A}_i=\bigcup _{j=1}^n{B}_j=S$ and each element of $S$ belongs to exactly $10$ of the $A_i$ 's and exactly $9$ of the $B_j$ 's, then the value of $n$ is

1. $115$
2. $35$
3. $45$
4. $25$

Q.

If $X=\{8^n-7n-1|nϵN\}$ and $Y=\{49n-49|nϵN\}$, then

1. $X\cap Y=\varphi$

2. $X\subset Y$

3. $Y\subset X$

4. $X=Y$

Q.

Let A = {x, y, z} and B = {1, 2}. Find the number of relations from to A to B.

Q. For each of the subsets of set $\{1,2,3,4,5,6,...,n\}$; let G be the greatest number. in that set
Then, find the sum of all such Gs.

1. ${1.2}^0+{2.2}^1+{3.2}^2+....+(n-1)2^{n-2}+n{.2}^{n-1}$
2. $n{.2}^n-2^{n-1}+1$
3. $n{.2}^n-2^n+1$
4. $n{.2}^n+2^{n-1}+1$

Q. 18. In how many ways can the letters of the word INTERMEDIATE be arranged so that all the vowels occupy even places?

Q. Number of ways the letters of word LETTER can be arranged such that vowels do not occur together

Q. If $X=\{8^n-7n-1|nϵN\}$ and $Y=\{49n-49|nϵN\}$, then

1. $X\subset Y$
2. $Y\subset X$
3. $X=Y$
4. $X\cap Y=\varphi$

Q. If $X=\{4^n-3n-1:n\in N\}\&Y=\left\{9\right(n-1):n\in N\}$, where $N$ is the set of natural numbers, then $X\cup Y$is equal to:

1. X
2. Y
3. N
4. Y - X