91Ó°ÊÓ

Perfect squares are numbers that can be expressed as the product of an integer with itself. For instance, when we multiply 2 by 2, the result is 4, which is a perfect square.

The set \(\{0,4,16,36,64,100, \ldots\}\) in the exercise consists of numbers where each one is a perfect square. These are the squares of the even natural numbers 0, 2, 4, 6, 8, 10, and so on. When describing perfect squares in set-builder notation, it is important to provide a general formula representing all numbers in the set.

For example, the perfect square of an even number can be expressed in a variable form as \(n^2\), where \(n\) is an integer. Thus, \(4\) is a perfect square because it can be written as \(2^2\), \(16\) as \(4^2\), and this pattern continues indefinitely.

Even natural numbers are all the non-negative integers that are divisible by 2 without leaving a remainder. They form a subset of the natural numbers, which includes the whole numbers starting from \(0\).

The sequence \(0, 2, 4, 6, 8, 10,\ldots\) lists the even natural numbers. When we refer to an even natural number \(n\), it can be mathematically expressed as \(2k\) where \(k\) is also a natural number. This shows that every even natural number is essentially two times some other natural number.

Considering the exercise, incorporating even natural numbers into set-builder notation requires identifying that only the squares of even natural numbers are included. This reflects the importance of understanding the specific properties of the numbers in a set.

Mathematical sets are collections of distinct objects or numbers, often represented within curly braces \(\{\}\). One way to precisely describe a set is by using set-builder notation, which specifies the properties that its elements must satisfy.

The general form of set-builder notation is \(\{ x | P(x) \}\), where \(x\) represents the elements of the set, and \(P(x)\) is a property or condition that the members must fulfill.

Set-Builder Notation for Perfect Squares of Even Numbers

In the context of our exercise, the set-builder notation for the set of perfect squares of even natural numbers is \[ \{ x: x = n^2, \text{ where } n \text{ is an even natural number} \} \]. This expression tells us that any number \(x\) in the set is the square of some even natural number \(n\). The property \(P(x)\) in this case is that \(x\) must be able to be written as \(n^2\), with \(n\) being an even natural number.