Universal Set
Definition: The universal set is the set of all elements under consideration.
Often times the universal set is hiding in the background. For example, consider the set \(\{1,2,3,4\}.\) The universal set may be the counting numbers \(\{1, 2, 3, \dots\},\) or all real numbers. So, it does not seem strange to say \(5 \not\in \{1, 2, 3, 4\}.\) However, what about my friend Joe? We could say Joe\(\not\in \{1, 2, 3, 4\},\) but we don't usually mix sets of people and sets of numbers this way.
It is also common to study certain universal sets in mathematics. The following are some common examples:
- Natural Numbers \(\mathbb{N}:\) The natural numbers, or counting numbers, is the set of numbers \(\{1, 2, 3, \dots\}\) It does not include 0 or negative numbers.
- Integers \(\mathbb{Z}:\) The integers extend the natural numbers to include 0 and negative whole numbers. It is the set \(\{0, 1, -1, 2, -2, 3, -3, \dots\}\)
- Rational Numbers \(\mathbb{Q}:\) The rational numbers include the integers and fractions of a whole over a whole. It is the set \(\{\frac{a}{b} : a, b \in \mathbb{Z}\}\)
- Real Numbers \(\mathbb{R}:\) The real numbers are difficult to define, and will be defined formally in a later lesson. The real numbers include all rational numbers and all decimals. For example, \(\pi = 3.1415\dots\) is a real number.
- Complex Numbers \(\mathbb{C}:\) The complex numbers include the imaginary number \(i = \sqrt{-1}.\) It is the set \(\{a + ib : a, b \in \mathbb{R}\}.\)
Empty Set
Definition: The Empty set is the set with no elements. It is written \(\emptyset\).
You could also write the empty set as \(\{\}\), but this is unusual.