Statements

From Department of Mathematics at UTSA
Revision as of 13:03, 27 September 2021 by Lila (talk | contribs)
Jump to navigation Jump to search

In mathematics, a statement (also known as a proposition) is a declarative sentence that is either true or false but not both. There must be no ambiguity as to whether the statement is true or false. The statement cannot be an opinion or vague sentence (for example, "chocolate tastes good" or "red is an ugly color") that does not have a set truth value. Some examples of basic statements and their truth values:

  • is an even number (TRUE)
  • Negative numbers are less than (TRUE)
  • (FALSE)
  • Quadrilaterals have five sides (FALSE)
  • For all real numbers , (TRUE)
  • All multiples of are even (FALSE)

Connectives

We can use connectives to create compound statements out of two or more basic statements, or to transform statements. Here are some common connectives and how they are used (let and be statements):

  • Conjunction: " and ", also denoted as "". A conjunctive statement is only true if both and are true, and false otherwise.
  • Disjunction: " or ", also denoted as "". A disjunctive statement is true if either or are true, and false only when both and are false.
  • Negation: "not ", also denoted as "". A negation reverses the truth value of the original statement (that is, if is true then is false, and vice versa).
  • Implication: "if , then " or " implies ", also denoted as "". An implication is true if and are both true, as well as when is false, regardless of the truth value of (that is, can be true or false when is false, and the implication will still be true). An implication is only false if is true and is false.
  • Biconditional: " if and only if ", also denoted as "". A biconditional is only true when both and " are true statements; that is, is true when and have the same truth value, and false when and have different truth values.

Truth table of some basic logical statements given statements and :

T T T T T T T F F
T F F T F T F F T
F T F T T F F T F
F F F F T T T T T

Resources