Question

State whether the given set if finite or infinite:
A = set of all triangles in a plane
B = set of all points on the circumference of a circle
C = set of all lines parallel to the y-axis
D = set of all leaves on a tree
E = set of all positive integers greater than 500
$F=\left\{ x\in R:0 < x <1 \right\}$
$G=\left\{ x\in Z:x <1 \right\}$
$H=\left\{ x\in Z:-15 < x < 15 \right\}$
$J=\left\{ x:x\in N\text{ }and\text{ }x\text{ }is\text{ }prime \right\}$
$K=\left\{ x:x\in N\text{ }and\text{ }x\text{ }is\text{ }odd \right\}$
L = set of all circles passing through the origin (0,0)

Answer 1

Hint: Finite sets means the number of elements in any set is countable like set $A=\left\{ A,P,L,E \right\}$. We have 4 elements in set A thus it is an example of finite set and infinite set means the number of element in set is not countable or uncountable, like set B $B=\left\{ x:x\in R:where\text{ }x\text{ }is\text{ }any\text{ }real\text{ }number \right\}$.

Complete step-by-step answer:

It is given in the question to state whether the given set is a finite or infinite set. So, basically finite set means the set having finite number of element or the number of element in set is countable, like set $X=\left\{ x\in N:x <5 \right\}$ which is $X=\left\{ 1,2,3,4 \right\}$ whereas, infinite set means the set having infinite number of element in set is uncountable, like set $Y=\left\{ y\in N:y\text{ }is\text{ }any\text{ }natural\text{ }number \right\}$ then set $Y=\left\{ 1,2,3,4,5,6,...... \right\}$.
Now in set 1, we have set of all triangles in a plane.

We can draw infinite triangles of different sides on the same plane thus, it is an infinite set.
Set B is set of all points on the circumference of circles.

We know that a circle is a locus of infinite points, that is the circumference of a circle has infinite points. Thus it is an infinite set.

Set C is a set of all lines parallel to the y-axis.

We can draw an infinite set parallel to the y-axis thus it is an infinite set.
Set D is the set of all leaves on a tree. We know that at a time, in a tree, we can count the total number of leaves. So, a tree has a countable number of leaves thus it is a finite set.
Set E is a set of all positive integers greater than 500. 500 is already a positive integer and we know that there are infinite positive integers greater than 500, thus it is an infinite set.
Set F is given as $F=\left\{ x\in R:0 < x <1 \right\}$, we know that there are infinite real numbers between two real numbers. So,we have infinite real numbers between 0 and 1, thus, F set is an infinite set.
Set G is given as $G=\left\{ x\in Z:x < 1 \right\}$, Z is an integer and x is less than 1 thus, we know that we have infinite integers less than 1, thus set G is an infinite set.
Set H is given as $H=\left\{ x\in Z:-15 < x <15 \right\}$, x is an integer but we have limit of x is -15 to 15 which mean it has fixed number of element in it thus it is a finite set.
Set J is given as $J=\left\{ x:x\in N\text{ }and\text{ }x\text{ }is\text{ }prime \right\}$, we know that we have an infinite prime number. Thus Set J is an infinite set.
Set K is given as $K=\left\{ x:x\in N\text{ }and\text{ }x\text{ }is\text{ }odd \right\}$, we have infinite odd numbers in set K, thus set K is an infinite set.
Set L is given as a set of all circles passing through the origin (0,0), we can draw an infinite circle passing through origin (0,0) thus set L is an infinite set.

Note: This question is a simple one, so students might try to guess and finish this question fast, but this guess work could lead the student into marking the sets as finite or infinite wrongly. So, it is recommended to do the conversions carefully, analyzing each set properly and then arriving at a conclusion.