An example of an infinite set in set builder form is {x | x ∈ ℕ}, which represents the set of all natural numbers.
Here's a breakdown:
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Set Builder Notation: Set builder notation provides a concise way to define a set by specifying the properties that its elements must satisfy. It generally follows the form: {x | P(x)}, where 'x' represents the elements of the set, and 'P(x)' is a predicate (a condition or property) that 'x' must satisfy to be included in the set.
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Infinite Set: An infinite set is a set with an unlimited number of elements. You can't count all the elements, as the process would never end.
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Natural Numbers (ℕ): Natural numbers are the set of positive integers, typically starting from 1 (i.e., 1, 2, 3, ...). Some definitions include 0. For the purpose of this example, we'll assume it starts from 1.
Therefore, {x | x ∈ ℕ} is read as "the set of all x such that x is an element of the natural numbers." Since the natural numbers are infinite, the resulting set is also infinite.
Other Examples:
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{x | x ∈ ℝ and 0 ≤ x ≤ 1}: This is the set of all real numbers between 0 and 1, inclusive. This is an infinite set because there are infinitely many real numbers between any two distinct real numbers.
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{x | x is an even integer}: This set contains all even integers (..., -4, -2, 0, 2, 4, ...), which is also infinite. It could be written as {x | x = 2n, n ∈ ℤ}, where ℤ is the set of all integers.
