The set of odd natural numbers divisible by 5 in set builder form is {x | x = 5(2n - 1), n ∈ ℕ}.
Let's break down why this is the correct set builder notation:
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{x | ... }: This indicates we are defining a set of elements 'x' such that they satisfy the condition that follows.
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x = 5(2n - 1): This is the key part. It specifies the rule for determining which numbers belong to the set.
- 5(...): Ensures that all elements are divisible by 5.
- (2n - 1): This expression always produces an odd number for any natural number 'n'. When n = 1, 2n - 1 = 1; when n = 2, 2n - 1 = 3; when n = 3, 2n - 1 = 5, and so on.
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n ∈ ℕ: This part clarifies that 'n' is a natural number. Natural numbers are positive integers (1, 2, 3, ...). This ensures we only include positive multiples of 5 in our set.
Examples:
- If n = 1, then x = 5(2(1) - 1) = 5(1) = 5
- If n = 2, then x = 5(2(2) - 1) = 5(3) = 15
- If n = 3, then x = 5(2(3) - 1) = 5(5) = 25
- If n = 4, then x = 5(2(4) - 1) = 5(7) = 35
And so on, generating the sequence 5, 15, 25, 35, 45... which are all odd natural numbers divisible by 5.
