Question

State whether each of the following set is finite or infinite:
A) The set of lines which are parallel to X-axis
B) The set of letters in English alphabets
C) The set of numbers which are multiple of 5.
D) The set of animals which are living on earth
E) The set of circles which are passing through the origin (0,0)

Answer 1

Hint: There are two types of sets called finite cells and infinite sets. The sets having the finite numbers of members or countable numbers of members are called finite sets. And sets having infinite numbers of members or uncountable numbers of members are called infinite sets. Here for each case we have to find the elements of sets. If the elements are countable then it will be a finite set otherwise infinite set.

Complete step-by-step answer:
Here we have to identify whether the given set is finite or not. Before this let’s learn what finite sets and infinite sets.
The sets having the finite numbers of members or countable numbers of members are called finite sets. And sets having infinite numbers of members or uncountable numbers of members are called infinite sets.
For example, set A: days in week.
Here $A = \left\{ {Sunday,monday,tuesday,wednesday,thursday,friday,saturday} \right\}$
Here members of set A is 7 which is countable/finite. So, set A is a finite set.
Say set $B = {2^n},n \in N$
Here
$B = \left\{ {2,4,8,16,32,64,.......} \right\}$
Here numbers of set B are uncountable. So set B is an infinite set.
Let’s check each set one by one.
A) The set of lines which are parallel to X-axis
Equation of line which is parallel to the X-axis is given by $y = cx$ where $c \in R$.
Here there are infinite possible values of c, so there will be infinite lines which are parallel to X-axis.
So, the given set of lines parallel to the x-axis is an infinite set.

B) The set of letters in English alphabets
In English there are a total 26 alphabets (A to Z).
So, set $A = \left\{ {A,B,C,D,E,....,X,Y,Z} \right\}$
Here the elements of set A are 26 which are countable.
So, a set of letters in English alphabets is a finite set.
C) The set of numbers which are multiple of 5.
Set A is numbers which are multiple of 5
So, $A = \left\{ {5n,n \in N} \right\}$
Simplifying, $A = \left\{ {5,10,15,20,...} \right\}$
Here there are infinite elements in set A, which are uncountable.
So a set of numbers which are multiple of 5 is an infinite set.
D) The set of animals which are living on earth.
The numbers of animals which are living on the earth are countable. Though it is difficult to calculate the number but somewhere it is a natural number. So elements of a set are countable. So, the set of animals which are living on earth are finite.
We can also call it a non-empty finite set.
E) The set of circles which are passing through the origin (0,0)
Equation of a circle whose center is (h,k) and passing through origin (0,0) is given by,
${(x - h)^2} + {(y - k)^2} = {h^2} + {k^2}$ And simplifying the equation we will get ${x^2} + {y^2} - 2hx - 2ky = 0$.
There are so many circles which will follow the above equation.
So the elements of the set of circles passing through origin will be uncountable.
So the set of circles passing through origin is an infinite set.

Note: If a set has a starting point and ending point both then it is a finite set. And if a set is having no end at any side or both sides then it is an infinite set. For a set, Cardinality represents the number of elements of a set. If the number of elements of set A is a, then the cardinality of set A is $n(A) = a$. If the set does not contain any elements it means it is an empty set ($A = \left\{ {} \right\}$ or $A = \emptyset $). Here elements are zero which is a finite number so the empty set is also a finite set.