Answer: (x, y) | (x, y) is a point on the Cartesian plane with x ∈ ℤ and y ∈ ℤ } is not an uncountable set.
Explanation:
The set (x, y) is a point on the Cartesian plane with x ∈ ℤ and y ∈ ℤ represents all the points on the Cartesian plane where both the x-coordinate and y-coordinate are integers.
To determine the cardinality (size) of this set, we need to consider whether it is finite or infinite, and whether it is countable or uncountable.
a) Uncountable set: An uncountable set is a set that is not in one-to-one correspondence with the set of natural numbers (ℕ). In other words, its elements cannot be counted one by one.
The set (x, y) is countable because we can establish a one-to-one correspondence between its elements and the set of natural numbers.
To see this, we can imagine listing the points on the Cartesian plane in a grid-like pattern, starting from the origin (0, 0) and moving outward. We can assign a unique natural number to each point based on its position in the grid. For example, the point (0, 0) can be assigned the number 1, (1, 0) can be assigned the number 2, (-1, 0) can be assigned the number 3, and so on.
Therefore, the set (x, y) is a point on the Cartesian plane with x ∈ ℤ and y ∈ ℤ is not an uncountable set.
b) Finite set: A finite set is a set that has a specific number of elements. In this case, the set includes an infinite number of points because there is no limit to the number of integer values for x and y.
Therefore, the set (x, y) is not a finite set.
c) Infinite set: An infinite set is a set that has an unlimited number of elements. As mentioned earlier, the set includes an infinite number of points because there is no limit to the number of integer values for x and y.
Therefore, the set (x, y) is a point on the Cartesian plane with x ∈ ℤ and y ∈ ℤ is an infinite set.
d) Nonexistent set: The set (x, y) is not a nonexistent set because it exists and has a well-defined definition.
To summarize, the set (x, y) is a point on the Cartesian plane with x ∈ ℤ and y ∈ ℤ is an infinite set and is not finite, uncountable, or nonexistent.