Question

What is the cardinality of the set of odd positive integers less than $10$?
A. $10$
B. $5$
C. $3$
D. $20$

Answer 1

It is a very simple problem where we just have to know the meaning of the word cardinality. It means the number of elements which are contained in the particular condition. Here we are given set of odd positive integers less than $10$ which are $\\{ 1,3,5,7,9\\}$
So we get that there are five numbers in the set which are satisfying the condition. So the cardinality of the given set is $5$

Complete step-by-step answer:
Here we are given to find the cardinality of the set of odd positive integers which are less than $10$
So we need to know what the meaning of the terms integers, cardinality and odd positive integers is. By their definition we will be easily able to solve the complete problem. So let us know their meaning=g one by one.
First of all cardinality is the number of elements that are contained in the set.
For example: If we are given to find the cardinality of the set given as $A = \\{ 0,1,5,6\\}$
So here we see that in the given set $A$ we have $4$ elements which are$0,1,5,6$. So we will simply say that the cardinality of the given set is $4$
So we are clear with one term now which is termed as cardinality of the set.
Now we must know what integers are. Integers include all the positive and negative numbers including zero which should not be written as the fractional components.
For example: $2,4,2025, - 1, - 6,0$ all are integers while $2.46,\dfrac{7}{{100}}$ are not.
Odd numbers means the numbers that are not divisible by $2$ completely whereas even are those that are completely divisible by $2$
For example: $1,3,5$ are all not divisible by $2$ so they are odd numbers while $2,6,8$ are all divisible by $2$ so are even numbers.
Now we need to find the cardinality of the set of odd positive integers less than $10$
We know the positive integers start from $1$ till infinity.
We need the positive integers that are less than $10$ which will be $\\{ 0,1,2,3,4,5,6,7,8,9\\}$
Out of these we need the odd positive integers which will be $\\{ 1,3,5,7,9\\}$
So we get our required set of odd positive integers which are less than $10$ as $\\{ 1,3,5,7,9\\}$
We need to find the cardinality of this set. So let us count the number of elements in this set which is $5$
So we get the cardinality of this set as $5$
Option B is correct.
Note: For solving such types of problems we must know the definitions of the important terms used in mathematics like what is the meaning of odd numbers, even numbers, integers, positive and negative integers is.
One main difference is to be known between positive and non-negative integers. Many students think them to be the same term but we have a difference. In positive integers we have all the positive natural numbers while in non-negative we have all the positive natural numbers including zero.