Set theory is closely related to logic. In this lesson, we make connections between concepts we have covered in set theory and concepts we have covered in logic.

\(\in\) and statements

Let \(U\) be the universal set, and let \(A\) be a set. For any \(x \in U,\) the claim \(x \in A\) is a statement. It is either true or false.

\(\cup\) and \(\vee\)

The statement \(x \in A \cup B\) means \(x \in A\) or \(x \in B.\) Converting the "or" to a logical operation, \[x \in A \cup B = (x \in A) \vee (x \in B)\]

\(\cap\) and \(\wedge\)

The statement \(x \in A \cap B\) means \(x \in A\) and \(x \in B.\) Converting the "and" to a logical operation, \[x \in A \cap B = (x \in A) \wedge (x \in B)\]

Complement and \(\neg\)

The statement \(x \in A^C\) means \(x\) is not in \(A.\) Converting the English to a logical statement, \[x \in A^C= \neg(x \in A)\]

\(\subset\) and \(\rightarrow\)

The statement \(A \subset B\) means that if \(x \in A\), then \(x \in B.\) Converting the English to a logical statement, \[A \subset B = (x \in A) \rightarrow (x \in B)\]

Universal sets and tautologies

Every element is in the universal set, \(U.\) So, \(x \in U\) is a tautology.

The empty set and contradictions

No element is in the empty set, \(\emptyset.\) So, \(x \in \emptyset\) is a contradiction.