Definition: The union of the sets \(A\) and \(B\) is the set \(\{c: c \in A \mbox{ or } c \in B\}\). The union of \(A\) and \(B\) is written \(A \cup B\).
Consider the following sets: \[A = \{1, 2\}, B=\{1, 4\}.\] The union of \(A\) and \(B\) is \[A \cup B = \{1, 2, 4\}.\] Even though 1 is in both sets, it is only listed once because sets are just lists of member elements. Remember that no element can be repeated.
The union of A and B is all of A and B combined.
Definition: The intersection of the sets \(A\) and \(B\) is the set \(\{c: c \in A \mbox{ and } c \in B\}\). The intersection of \(A\) and \(B\) is written \(A \cap B\).
Consider the following sets: \[A = \{1, 2, 3\}, B=\{1, 3, 4\}.\] The intersection of \(A\) and \(B\) is \[A \cap B = \{1, 3\}.\] The only elements in both \(A\) and \(B\) are 1 and 3.
The intersection of \(A\) and \(B\) is the part of \(A\) and \(B\) that are the same.
Definition: The complement of the set \(A\) is the set of all elements in the universal set that are not in \(A.\) If the universal set is \(U\), then the complement of \(A\) is the set \(\{x \in U: c \not\in A\}\). The complement of \(A\) is written \(A^C\).
Suppose we are studying the set of numbers \(\{1, 2, 3, 4, 5\},\) so the universal set is \(U = \{1, 2, 3, 4, 5\}.\) Let \(A\) be the set \(\{2,3,5\}.\) Then \(A^C = \{1,4\}.\)
As a more interesting example, let \(U = \mathbb{N}\) be the set of natural numbers. Let \(E\) be the set of even numbers, \(E = \{2, 4, 6, \dots\}.\) Then, \(E^C\) is the set of odd numbers. The complement is the set \(E^C = \{1, 3, 5, \dots\}.\)
The compliment of \(A\) is the part of the universe that is not in \(A.\)
The universal set \(U\) is everything in the blue square and red circle.
The red circle is the set \(A.\)
The blue area outside of the circle is the copmliment of \(A.\)