Question

5- Use set builder notation to define the set of points ( x−y coordinates / ordered pairs) on the curve defined by y=x^2 where x and y are integers.

Answer 1

The set of points on the curve y = x², with integer coordinates, in set builder notation is (x, x²) . This includes any pair where the first element is an integer and the second element is the square of the first.

Step-by-step explanation:

To define the set of points on the curve described by the equation y = x² where x and y are integers, we use set builder notation. Set builder notation uses a descriptive phrase or a mathematical property to describe the members of a set. In this case, we have the specific property that our y values are the squares of our x values, and both x and y must be integers.

The set of points (or ordered pairs) that lie on the curve can be expressed in set builder notation as:

x is an integer

This means that the set consists of all possible ordered pairs where the first element is an integer, and the second element (y) is the square of the first element (x). For example, (2, 4), (-3, 9), and (0, 0) are all possible members of this set since they satisfy the given relationship and both elements of the pairs are integers.

Answer 2

The set of points (x, y coordinates) on the curve defined by
`\(y=x^2\)`where x and y are integers can be expressed in set builder notation as
`\(\{(x, x^2) \mid x \in \mathbb{Z}\}\).`

Step-by-step explanation:

To define the set of points on the curve
`\(y=x^2\)` where both x and y are integers using set builder notation, we represent the coordinates as ordered pairs
`(x, x^2)`, where x belongs to the set of integers
`\(\mathbb{Z}\)`. In set builder notation, this set can be expressed as
`\(\{(x, x^2) \mid x \in \mathbb{Z}\}\)`, indicating that x takes on integer values, and the corresponding y values are determined by the expression
`\(x^2\)`.

For example, if x is 1, the corresponding y value is
`1^2` = 1, resulting in the ordered pair (1, 1). Similarly, if x is -2, the corresponding y value is
`(-2)^2 = 4`, giving us the ordered pair (-2, 4). This notation succinctly captures all possible ordered pairs on the curve for integer values of x.

The set builder notation is a concise and precise way to describe mathematical sets, and in this case, it precisely defines the set of points on the given curve with integer coordinates. It encapsulates the relationship between x and y as dictated by the equation
`\(y=x^2\)`, with x restricted to integer values.