Having identified statements and primitive statements, the next step is to change or combine statements. The first logical connective is negation, which reverses the truth value of a statement. Negation is also the simplest way to form a compound statement from one given statement. In this lesson, negation and compound statements prepare the notation used throughout truth tables and symbolic logic. :::definition[Negation] Let $p$ be a statement. The negation of $p$ is the statement that asserts the opposite truth value of $p$. It is denoted by $\neg p$ and is read as not $p$. ::: Negation does not mean that a sentence becomes vague or uncertain. It means that the truth value is reversed. If $p$ is true, then $\neg p$ is false. If $p$ is false, then $\neg p$ is true. Thus, negation is a truth-value operation, not merely a grammatical change in wording. :::example[Negation of a Statement] Let $$\begin{equation*} p:\text{ The lecture begins at }10\text{ a.m.} \end{equation*}$$ Then $$\begin{equation*} \neg p:\text{ The lecture does not begin at }10\text{ a.m.} \end{equation*}$$ Thus, the negation changes the truth value of the original statement. ::: :::definition[Truth Table for Negation] Let $p$ be a statement. The truth table for $\neg p$ is $$\begin{equation*} \begin{array}{c|c} p & \neg p\\ \hline 0 & 1\\ 1 & 0 \end{array} \end{equation*}$$ Here $1$ denotes true and $0$ denotes false. ::: :::definition[Compound Statement] A compound statement is a statement formed from one or more statements by using logical connectives. ::: :::definition[Basic Logical Connectives] The basic logical connectives are $\neg$, $\land$, $\lor$, $\to$, and $\leftrightarrow$. They represent negation, conjunction, disjunction, implication, and biconditional respectively. ::: Negation is a unary connective because it is applied to one statement. Conjunction, disjunction, implication, and biconditional are binary connectives because each joins two statements. A common mistake is to think that every compound statement must contain two primitive statements. The statement $\neg p$ is already compound because it is formed from $p$ using the connective $\neg$. :::example[Compound Statements] Let $$\begin{align*} p &: \text{The student attends class.}\\ q &: \text{The student submits the assignment.} \end{align*}$$ Then $\neg p$ means The student does not attend class. The statement $p\land q$ means The student attends class and submits the assignment. The statement $p\to q$ means If the student attends class, then the student submits the assignment. Each of these is a compound statement. ::: :::solved-problem Let $p:$ The exam is scheduled on Monday, and $q:$ The classroom is available. Write the symbolic forms of the following statements: (1) The exam is not scheduled on Monday. (2) The exam is scheduled on Monday and the classroom is available. (3) If the exam is scheduled on Monday, then the classroom is available. (4) The exam is scheduled on Monday if and only if the classroom is available. ::: :::solution Let $p$ denote The exam is scheduled on Monday and let $q$ denote The classroom is available. [1] The exam is not scheduled on Monday is the negation of $p$. Hence, its symbolic form is $\neg p$. [2] The word and is represented by conjunction. Hence, the symbolic form is $p\land q$. [3] The phrase if $p$, then $q$ is represented by implication. Hence, the symbolic form is $p\to q$. [4] The phrase if and only if is represented by biconditional. Hence, the symbolic form is $p\leftrightarrow q$. ::: :::solved-problem Let $p$ be true and $q$ be false. Determine the truth values of $\neg p$, $\neg q$, and $\neg(\neg q)$. ::: :::solution Let $p=1$ and $q=0$. Since negation reverses truth value, $$\begin{align*} \neg p &= 0,\\ \neg q &= 1,\\ \neg(\neg q) &= \neg 1\\ &=0. \end{align*}$$ Hence, $\neg p$ is false, $\neg q$ is true, and $\neg(\neg q)$ is false. ::: :::exercise Let $p:$ The switch is on, and $q:$ The battery is charged. Write symbolic forms for: (1) The switch is not on. (2) The battery is not charged. (3) The switch is on and the battery is charged. (4) It is not the case that the battery is charged. ::: :::answer (1) $\neg p$. (2) $\neg q$. (3) $p\land q$. (4) $\neg q$. ::: :::faq[Frequently Asked Questions] Q: Is $\neg p$ a primitive statement? A: No. It is formed from $p$ by using negation, so it is a compound statement. Q: Does negation always use the word not? A: In English, negation may be expressed in several ways. In symbolic logic, all such truth reversals are represented by $\neg$. Q: Can a compound statement contain only one primitive statement? A: Yes. The statement $\neg p$ contains one primitive statement and one connective. ::: :::call-to-action[Strengthen Negation Skills] subtitle: Practise reversing truth values before moving to conjunction and disjunction, where two statements are combined. button: Next Lesson | /discrete-mathematics/mathematical-logic/basic-logical-connectives-and-truth-tables/conjunction-and-disjunction button-ghost: Previous Lesson | /discrete-mathematics/mathematical-logic/basic-logical-connectives-and-truth-tables/statements-and-primitive-statements :::