Subset

Q.

Prove that $6+\surd 2$ is an irrational number.

Q.

Let A, B , C be three sets . If A is subset of B abd B belnngs to c, is it true that A belongs to C ? If not given qn example

Q.

Represents the irrational numbers on the number line.

Q.

Is $\log 2$ rational or irrational? Justify your answer.

Q.

Prove that $5\surd 2$ is irrational.

Q.

The number of subsets of a set containing n element is

1. $2^n$

2. $2^n-1$

3. $n^2$

4. n

Q. Let $A,B$ and $C$ be three sets. If $A\in B$ and $B\subset C$, is it true that $A\subset C$ ?. If not, give an example.

Q.

What is an irrational number?

Q.

Let $R=\left\{\left(3,3\right),\left(6,6\right),\left(9,9\right),\left(12,12\right),\left(6,12\right),\left(3,9\right),\left(3,12\right),\left(3,6\right)\right\}$ be a relation on set $A=\left\{3,6,9,12\right\}$. The relation is

1. An equivalence relation

2. Reflexive and symmetric only

3. Reflexive and transitive only

4. Reflexive only

Q. If $A$ and $B$ are finite sets and $A\subset B,$ then

1. $n(A\cup B)=n\left(A\right)$
2. $n(A\cap B)=n\left(B\right)$
3. $n(A\cup B)=n\left(B\right)$
4. $n(A\cap B)=\varphi$

Q. There are two factories located one at place P and the other at place Q. From these locations, a certain commodity is to be delivered to each of the three depots situated at A, B and C. The weekly requirements of the depots are respectively 5, 5 and 4 units of the commodity while the production capacity of the factories at P and Q are respectively 8 and 6 units. The cost of transportation per unit is given below:

From \ To	Cost (in ₹)
	A	B	C
P	160	100	150
Q	100	120	100

How many units should be transported from each factory to each depot in order that the transportation cost is minimum. What will be the minimum transportation cost?

Q. If the total number of $m$-element subsets of the set $A=\{a_1,a_2,...,a_n\}$ is $k$ times the number of $m$ element subsets containing $a_4$ then $n$ is

1. $(m+1)k$
2. $(m-1)k$
3. $mk$
4. $(m+2)k$

Q.

Let A = {$\varphi$ {$\varphi$}, 1, {1, $\varphi$}, 2}. Which of the following are true ?

(i) $\varphi ϵA$ (ii) {$\varphi$} $ϵ$ A

(iii) {1} $ϵ$ A (iv) {2, $\varphi$} $\subset$ A

(v) 2 $\subset$ A (vi) {2, {1}} $⊈$ A

(vii) {{2}, {1}}, $⊈$ A

(viii) { $\varphi$, {$\varphi$} , {1, $\varphi$}} $\subset$ A

(ix) {{ $\varphi$}} $\subset$ A.

Q.

If A is any set , prove that :

$A⊈\varphi \iff A=\varphi$.

Q.

Which of the following statements are correct ? Write a correct form of each of the incorrect statements.

(i) a $\subset$ {a, b, c}

(ii) {a} $ϵ$ {a, b, c}

(iii) a $ϵ$ {(a), b}

(iv) {a} $\subset$ {(a), b}

(v) {b, c} $\subset$ {a, {b, c}}

(vi) {a, b} $\subset$ {a, {b, c}}

(vii) $\varphi ϵ$ {a, b}

(viii) $\varphi \subset$ {a, b, c}

(ix) {x : x +3 = 3} = $\varphi$

Q.

Let $A=\left\{1,2\right\}$, $B=\left\{3,4\right\}$. Then, the number of subsets of $A\times B$ is

1. $4$

2. $8$

3. $18$

4. $16$

Q.

If three positive real numbers $\mathrm{a},\mathrm{b},\mathrm{c}$ are in A. P. and if $\mathrm{abc}=64$ then the minimum value of $\mathrm{b}$ is

1. $6$

2. $5$

3. $4$

4. $3$

Q.

Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8 }}. Determine whch of the following is true or false :

(i) 1 $ϵ$ A (ii) {1, 2, 3} $\subset$ A (iii) {6, 7, 8} $ϵ$ A (iv) {{4, 5}} $\subset$ A (v) $\varphi$ $ϵ$ A

Q. Out of the four alternatives, choose the one which expresses the sentence below in passive voice.

The manager could not accept the union leader's proposals.

1. The union leader's proposals could not have been accepted by the manager.
2. The union leader's proposals could not had been accepted by the manager.
3. The union leader's proposals could not accepted by the manager.
4. The union leader's proposals could not be accepted by the manager.

Q. Which set is the subset of all given sets

1. $\{1,2,3,\mathrm{4...}\}$
2. $\left\{1\right\}$
3. $\left\{0\right\}$
4. $\left\{\right\}$

Q. Let $A=\{2,4,6\}$ and $B=\{x:x\text{is an even natural number less than}8\}$, then which of the following is/are correct?

1. $A\subset B$
2. $B\subset A$
3. $A\ne B$
4. $n\left(A\right)<n\left(B\right)$

Q.

If A and B are two sets such that $AsubsetB,$ then find :

(i) $A\cap B$ (ii) $A\cup B$

Q.

Let A = {a, b, {c, d}, e}. Which of the following are false and why ?

(i) {c, d} $\subset$ A (ii) {c, d} $ϵ$ A (iii){{c, d}$ϵ$ A }

(iv) a $ϵ$ A (v) a $\subset$ A (vi) {a, b, e} $\subset$ A

(vii) {a, b, e} $ϵ$ A (viii) {a, b, c} $ϵ$ A (ix) $\varphi$$ϵ$ A (x) { $\varphi$ } $\subset$ A

Q.

Write the set of all vowels in the English alphabet which precede q.

Q.

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

Q.

Write which of the following statements are true ? Justify your answer.

(i) The set of all intergers is contained in the set of all rational numbers.

(ii) The set of all crows is contained in the set of all birds.

(iii) The set of all rectangles is contained in the set of all squares.

(iv) The set of all real numbers is contained in the set of all complex numbers.

(v) The sets P = {a} and B = {{a}} are equal.

(vi) The sets A = {x : x is a letter of the word "LITTLE"} and B = {x : x is a letter of the word "TITLE"} are equal.

Q. The number of unordered pairs $(A,B)$ of subsets of the sets $S=\{1,2,3,4,5,6\}$ such that $A\cap B=\varphi$ and $A\cup B=S$ is

1. $32$
2. $64$
3. $128$
4. $63$

Q.

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

Q.

Write down all possible subsets of each of the following sets :

(i) {a} (ii) {0, 1}

(iii) {a, b, c} (iv) {1, {1}}

(v) {$\varphi$}

Q. If $U=\{1,3,5,7,9,11,13\}$, then which of the following is/are the subsets of $U$?

1. $A=\left\{0\right\}$
2. $B=\{1,9,5,13\}$
3. $C=\{3,11,7\}$
4. $D=\{2,3,4,5\}$