Statements

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:

• $2$ is an even number (TRUE)
• Negative numbers are less than $0$ (TRUE)
• $3+4=6$ (FALSE)
• Quadrilaterals have five sides (FALSE)
• For all real numbers $n$, $n^{2}\geq 0$ (TRUE)
• All multiples of $3$ 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 $P$ and $Q$ be statements):

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

Truth table of some basic logical statements given statements $P$ and $Q$:

$P$	$Q$	$P\land Q$	$P\lor Q$	$P\Rightarrow Q$	$Q\Rightarrow P$	$P\Leftrightarrow Q$	$\neg P$	$\neg Q$
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

• Course Textbook, pages 4-25
• Statements and Logical Operators, OpenStax