Question

Write the set of all positive integers whose cube is odd.

Answer 1

Hint: To find the set of all positive integers whose cube is odd, first, we will determine the nature of the cube of the odd and even positive integers and then we will create a set of those numbers only whose cube is odd.

Complete step-by-step solution -
Before solving the question, we must first know what are positive odd integers. A positive odd integer is a natural number which is of the form $\left( {2n - 1} \right)$ where n is a positive whole number. In other words, it is not divisible by 2. Thus the positive odd integers are 1,3,5,7,9,11,13,……
Now, we will find the cube of the natural numbers and we will determine their nature. Thus, we have:
Cube of 1 $ = {\left( 1 \right)^3} = 1 \times 1 \times 1 = 1$
Cube of 2$ = {\left( 2 \right)^3} = 2 \times 2 \times 2 = 8$
Cube of 3 $ = {\left( 3 \right)^3} = 3 \times 3 \times 3 = 27$
Cube of 4 $ = {\left( 4 \right)^3} = 4 \times 4 \times 4 = 64$
Cube of 5 $ = {\left( 5 \right)^3} = 5 \times 5 \times 5 = 125$
Cube of 6 $ = {\left( 6 \right)^3} = 6 \times 6 \times 6 = 216$
Here, we can see that the cubes of 1,3 and 5 are positive odd integers while cubes of 2,4 and 6 are even integers. Similarly, we can say that the cubes of 7,9,13,15 will also be odd. Now, we have to create a set of all the positive integers whose cube is odd. Before we create a set, we must know what a set is. A set is a collection of well-defined and distinct objects. Let the set be named ‘S’ then:
$S = \left\{ {1,3,5,7,9,11,13........} \right\}$
We cannot write all elements in a set as it is an infinite set.

Note: This question might be a little confusing. Here, we do not have to make a set of odd cubes of positive integers but we have to create a set of those numbers whose cube is an odd positive number.