Hi I not that sure how to write the sets in roster notation especially with the ones with x^2. Thanks for the help!

Roster notation, also known as list notation, is a way of expressing the elements of a set by explicitly listing each element. In roster notation, we use curly braces { } to enclose the elements of the set, and commas to separate the elements.

[tex]\hrulefill[/tex]

Part (a)

[tex]\{ x \in \mathbb{N} \,|\, 15 \leq x^2 < 70 \}[/tex]

The notation [tex]x \in \mathbb{N}[/tex] means "x belongs to the set of natural numbers."

Natural numbers are all positive integers from 1 to infinity.

The inequality 15 ≤ x² < 70 means that we need to find the natural numbers whose squares are greater than or equal to 15 and less than 70.

The first few natural numbers (x) are:

[tex]x = 1, 2, 3, 4, 5, 6, 7, 8, ...[/tex]

So, the squares of the first few natural numbers are:

[tex]x^2 = 1, 4, 9, 16, 25, 36, 49, 64, ...[/tex]

The squares (x²) that satisfy the inequality are 16, 25, 36, 49 and 64, so the corresponding natural numbers (x) are 4, 5, 6, 7 and 8.

Therefore, the set in roster notation is {4, 5, 6, 7, 8}.

[tex]\hrulefill[/tex]

Part (b)

[tex]\{ x^2 \,|\, x \in \mathbb{N} \, \text{and} \, 15 \leq x^2 < 70 \}[/tex]

The given set notation means:

The set of all square numbers (x²) such that x belongs to the set of natural numbers AND x² must be greater than or equal to 15 and less than 70.

As we have already found the squares of natural numbers that satisfy the inequality 15 ≤ x² < 70 in part (a), then the set in roster notation is {16, 25, 36, 49, 64}.

[tex]\hrulefill[/tex]

Part (c)

[tex]\{ (x, y, x) \,|\, x \in \{A, C, U, G\}, y \in \{A, C\}[/tex]

This set represents ordered triples (x, y, x) where the first and third elements are the same (x), and x comes from the set {A, C, U, G}, while y comes from the set {A, C}.

The roster notation is simply the list of all the possible triples:

{(A, A, A), (A, C, A), (C, A, C), (C, C, C), (U, A, U), (U, C, U), (G, A, G), (G, C, G)}