Finite and infinite sets - class-IX

Since the number of boys and girls in a class are always finite.
Therefore, set G and H are finite sets.

C is an infinite set as there are many numbers less than $ - 3 $

Since the numbers of animals on the earth are countable(limited).
Therefore, set of all the animals on the earth is "finite" set.
Option A is correct.

A non empty set has atleast one element.

From the given options we see that the SET C has one element which is $ 2 $

(2), (3), and (4) are finite sets.

The set of positive integers is never ending. There is no such defined largest integer. Hence the set is infinite

Definition: A finite set is a set which has fixed or finite number of elements.
$A=$ Set of animals on the earth is a finite set because number of animals on the earth is large in number, but it is finite/fixed.

$(i)(x-1)(x-2)=0$

The set ${ (x,y):{ x }^{ 2 }+{ y }^{ 2 }\le 1\le x+y,\ \ x,y\in R} $ consists of all the points in the first quadrant which lie inside the circle ${ x }^{ 2 }+{ y }^{ 2 }=1$ and above the line $x+y=1$ .So, it is not a finite set.
Option $A$ is correct.

The intersection of an infinite set may be finite.
Example:
$A={x:x\in N; x>2}$
$B={x:x\in I; x<5}$
Here, Both $A$ and $B$ are infinite but its intersection are finite.

Definition: A set having infinite number of elements is known as infinite set.
 ${x:x\in R:1\le x\le 3}$ is a infinite set because since $x\in R$, there are infinte number of real numbers lie in between two numbers.

Given the set contains $n$ elements.

A will be an infinite set as we have many multiples of 3 such has 3, 6, 9, 12 and so on

Set D will be infinite as there will be infinite powers of 2 since n can be any natural number from 1 to infinity.

B is a finite set as 13 has only two factors which are 1, and 13.

We can have many fractions between the numbers 3 and 4.
Hence, this set is an infinite set

If A and B are finite sets with the same number of elements (n(A) = n(B)) but A is not equal to B, then the intersection A intersect B must contain fewer elements than A. If it contained n(A) elements, then A would have to be a subset of B, and since they have the same size, A would equal B.

We have to identify the type of set.

Given $A={x|x\in N, 2 \leq x \leq 3 }$

               $={2,3}$ which is a finite set.

Therefore $A$ is a finite set.

$S$ is given by $S={x:x\in N: x\le15}$
$S={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}$
$\therefore n(S)=15$
The power set is set of all the possible subsets of $S.$
Number of elements in power set $=$ Total number of subsets
So, cardinality of power set $S=2^{15}=32768$

Countably Infinite: Set of natural numbers are countably infinite because one can count natural number one after other easily.
Uncountably infinite: Set of real numbers are uncountably infinite because real numbers cannot be counted.

$(i)A={x:x\in Z: x^2 $ is even $}$
$A={...,-6,-4,-2,0,2,4,6,...}$ which is an infinite set.
$(ii)B={x:x\in R:-4<x<-2}$
There will be infinite real numbers any two numbers so, it is an infinite set.

$(i)$The set of lines which are parallel to x-axis is an infinite set because line parallel to x-axis are infinite in number.
$(ii)$The set of animals living on the earth is a finite set because the number of animals living on the earth is finite (although it is quite a big number)
$(iii)$ The set of numbers which are multiple of $5$ is an infinite numbers multiples of $5$ are infinite in number.
$(iv)$ The set of the circles passing through the origin $(0,0)$ is an infinite set because infinite number of circles can pass through the origin.

$(i)$ The sets of months in a year is a finite set because it has $12$ elements.
$(ii){1,2,3,....}$ is an infinite set as it has infinite number of elements.
$(iii){1,2,3,...,99,100}$ is a finite set as it has number from $1$ to $100$ which is finite in number.
$(iv)$ The set of positive integers greater than $100$ is an infinite set because positive integers greater than $100$ are infinite in number. 

Given that:
$A={a,{b}},$
 $P(A)={\phi,a,{b},{a,{b}}}$

$(i)A={x:x\in Z: x^2 $ is even $}$
$A={...,-3,-1-1,3,...}$ which is an infinite set.
$(ii)B={x:x\in R:-2<x<-4}$
$B={...,-14,-13,-12,-11}$ so it is an infinite set.