Finite and Infinite Set

Q. Which of the following is an infinite set?

1. The set of human beings on the earth
2. The set of water drops in a glass of water
3. The set of trees in a forest
4. The set of all prime numbers

Q.

Which of the following statements are true ? Give reason to support your answer.

(i) For any two sets A and B either A $\subseteq$ B or B $\subseteq$ A.

(ii) Every subset of an infinite set is infinite.

(iii) Every subset of a finite set is finite.

(iv) Every set has a proper subset.

(v) {a, b, a, b, a, b....} is an infinite set.

(vi) {a, b, c} and {1, 2, 3} are equivalent sets.

(vii) A set can have infinitely many subsets.

Q. Consider the function, $f\left(x\right)=x+1,0\le x\le 2,$ then $fof\left(x\right)$ is

1. $x+2,0\le x\le 1$
2. $x+2,0\le x\le 2$
3. $x+2,0\le x\le 4$
4. $x+2,0\le x\le 3$

Q. Which of the following is a finite set ?

1. $A=\{x:x\in N\text{and}x\text{is a prime number}\}$
2. $B=\{x:x\text{is a real number between}1\text{and}10\}$
3. $C=\{x:x\text{is a perfect square number between}0\text{and}99\}$
4. $D=\{x:x\text{is an integer less than}20\}$

Q.

Which of the following sets are finite and which are infinite ?

(i) Set of concentric circles in a plane.

(ii) Set of letter of the English Alphabets.

(iii) {$xϵN;x>5$}

(iv) {$xϵN;x>200$ }

(v) {$xϵZ;x<5$ }

(vi) {$xϵR;0<x<1$ }.

Q. The most appropriate option for the following sets
$A$ is set of letters of the word 'NARAYANA'
$B$ is the set of letters of the word 'ARANYA'

1. $A$ and $B$ are equivalent sets
2. $A$ and $B$ are equal sets
3. $A$ and $B$ are similar sets
4. $n\left(A\right)>n\left(B\right)$

Q.

Which of the following is a finite set?

1. Set of all 4 digit prime number

2. Set of all positive integers which are multiple of 5

3. The set of positive integers greater than 100.

4. Set of points on a line.

Q. State whether each of the following set is finite or infinite:
$\left(i\right)$ The set of lines which are parallel to the $x-$axis
$\left(ii\right)$ The set of letters in the English alphabet
$\left(iii\right)$ The set of numbers which are multiple of $5$
$\left(iv\right)$ The set of animals living on the earth
$\left(v\right)$ The set of circles passing through the origin $(0,0)$

Q. 37. If the plane 3x+y+2z+6=0 is parallel to the line 3x-1/2b =3-y=z-1/a, then the value of 3a+3b

Q.

Evaluate the definite integrals.
${\int }_0^{\pi }(si{n}^2\frac{x}{2}-co{s}^2\frac{x}{2})dx.$

Q.

Let $U$ be the universal set and$A\cup B\cup C=U.$ Then $\left\{\right(A-B)\cup (B-C)\cup (C-A)\}$ is equal to

1. $A\cup B\cup C$

2. $A\cup (B\cap C)$

3. $A\cap B\cap C$

4. $A\cap (B\cup C)$

Q. State, whether each of the following sets is a finite set or an infinite set:

$\text{{}x:x=3n-2,n\in Z,n\le 8\text{}}$

Q.

State, whether the following statement is true? Give reason to support your answer.

(iii) Every subset of a finite set is finite.

Q. Which of the following is/are singleton sets?

1. $A=\{x:x\text{is H.C.F. of}12\text{and}18\}$
2. $B=\{x:x\text{is a common prime factor of}16\text{and}27\}$
3. $C=\{x:x\text{is a prime factor of}35\}$
4. $D=\{x:x\text{is L.C.M. of}6\text{and}11\}$

Q. Answer, whether the following statements are true or false. Give reasons
The set of even natural numbers less than 21 and the set of odd natural numbers less than 21 are equivalent sets.

Q.

Evaluate the definite integrals.
${\int }_0^{\frac{\pi }{2}}cos2xdx.$

Q.

Identify the finite and infinite sets and find the cardinal numbers of the finite sets$.$

$E=\{x:x<1$and $x\in N\}$

Q.

DEFNITION - A set which is empty or consists of a different number of element is called a finite set? THIS DEFNITION IS WRITTEN IN NCERT BOOK. WHY WE USE THE WORD EMPTY THEIR?

Q. If $A=\{b,e,f,g\}$ and $B=\{c,e,g,h\}$, then verify the commutative property of (i) union of set, (ii) intersection of sets

Q. Find the cardinal number of the following sets:
$A_4=\{b:b\in Zand-7<3b-1\le 2\}$

Q.

Observe the series $2,4,6,8,10$ … and write the generalised formula.

Q. For each set, given below, state whether it is finite set, infinite set or the null set

vii. {squares of natural numbers}

Q. Give one example each for a finite set and an infinite set.

Q. Mark the correct alternative in the following question:

Let A = {1, 2, ... , n} and B = {a, b}. Then the number of subjections from A into B is

(a) ⁿP₂ (b) 2ⁿ $-$ 2 (c) 2ⁿ $-$ 1 (d) ⁿC₂

Q. Answer, whether the following statements are true or false. Give reasons

ii If $E=\text{{factor of 16}}$ and $F=\text{{factor of 20}}$, then $E=F$.

Q. State, whether the following pairs of sets are equivalent or not:

Set of integers and set of natural numbers.

Q. State, whether the following pairs of sets are equivalent or not:

P ={5, 6, 7, 8} and M={$x:x\in Wandx\le 4$}

Q.

If $A$and $B$are two equivalence relations defined on a set $C$, then

1. $A\cap B$ is an equivalence relation

2. $A\cap B$is not an equivalence relation

3. $A\cup B$is an equivalence relation

4. $A\cup B$is not an equivalence relation

Q. If $A=\{1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25}\}$, then write $A$ in Set-builder form.

Q.

By using properties of definite integrals, evaluate the integrals
${\int }_0^{\frac{\pi }{2}}co{s}^{2}dx.$