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State whether the given statement is true or false. The set of squares of integers and the set of whole numbers are equal sets.

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Last updated date: 25th Jul 2024
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Answer
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Hint: Here, in the question, a statement is given that “The set of squares of integers and the set of whole numbers are equal sets” and we are asked to state whether this statement is true or false. We will first understand the integers and whole numbers separately and then build their set to get the desired result.

Complete step-by-step solution:
Integers: Integers are all the numbers but not the fractional ones. In simple words, integers include all the natural numbers, negatives of natural numbers and zero but no fraction allowed.
Whole numbers: Whole numbers are all the natural numbers including zero.
Let \[Z\] be the set of integers,
Therefore, \[Z = \left\{ { - \infty , \ldots , - 3, - 2, - 1,0,1,2,3, \ldots ,\infty } \right\}\]
Now, if we write the set of squares of all integers (let’s say \[A\]), it will be written as,
\[A = \left\{ {0,1,4,9,16, \ldots } \right\}\]
The set of whole numbers (let’s say \[B\]), will be written as,
\[B = \left\{ {0,1,2,3,4,5, \ldots ,\infty } \right\}\]
Clearly, we can see that the sets \[A\] and \[B\] are not equal as we don’t have the elements 2,3,6... in set A.
Hence, the given statement “The set of squares of integers and the set of whole numbers are equal sets” is false.

Note: Given two sets \[A\] and \[B\], if every element of \[A\] is also an element of \[B\] and if every element of \[B\] is also an element of \[A\], then the sets \[A\] and \[B\] are said to be equal. Clearly, the two sets should have exactly the same elements. In the above question, every element of set \[A\] is also the element of set \[B\] but every element of \[B\] is not necessarily the element of \[A\]. Therefore, two sets are not equal.