In set theory, Z represents the set of all integers.
Integers include all whole numbers (positive, negative, and zero). This means Z encompasses numbers like -3, -2, -1, 0, 1, 2, 3, and so on, extending infinitely in both positive and negative directions.
Here's a breakdown:
- Symbol: Z (often written as ℤ)
- Definition: The set of all integers.
- Elements: {..., -3, -2, -1, 0, 1, 2, 3, ...}
- Relationship to other sets:
- Z contains the set of natural numbers (N).
- Z is a subset of the set of rational numbers (Q).
- Z is a subset of the set of real numbers (R).
- Z is a superset of the set of whole numbers (W).
- Z is not a subset of the set of imaginary numbers.
Examples:
- 5 ∈ Z (5 is an element of Z, meaning 5 is an integer)
- -10 ∈ Z (-10 is an element of Z, meaning -10 is an integer)
- 0 ∈ Z (0 is an element of Z, meaning 0 is an integer)
- 1.5 ∉ Z (1.5 is not an element of Z, meaning 1.5 is not an integer)
- √2 ∉ Z (√2 is not an element of Z, meaning √2 is not an integer)
In summary, Z is a fundamental set in mathematics that includes all whole numbers and their negative counterparts.
