Introduction to Logic

Lessons

Senior 4 · Mathematics

Introduction to Logic

5 lessons

Lesson 1 of 5

Statements, connectives and truth tables

25 min

"And" is not the same as "or"

A school notice says: "Visitors may enter if they sign the register and show an identity card." A visitor who shows a card but does not sign is turned away. Had the notice said "or", the same visitor would walk in. In everyday talk such words are used loosely; in mathematics, computing and law their meaning must be exact, because a proof, a program or a contract depends on it. Logic is the study of exact reasoning: which sentences can be true or false, how they combine, and when a conclusion really follows from what is given (an argument). This lesson starts with the building blocks.

Definition

Statement (proposition): a declarative sentence that is either true or false, but not both. Its truth value is written T (true) or F (false); some books write 1 and 0. Statements are named by small letters $p, q, r, \ldots$

Argument: a list of statements, the premises, followed by a statement claimed to follow from them, the conclusion. Lesson 5 shows how to test whether an argument is valid.

Examples: "91 is divisible by 7" is a true statement ( $91 = 7 \times 13$ ); " $2 + 3 = 6$ " is a false statement; "Close the door" is not a statement.

Statement or not?

Sentence	Statement?	Reason
Kigali is the capital of Rwanda.	Yes (T)	declarative, and true
51 is a prime number.	Yes (F)	declarative, and false: $51 = 3 \times 17$
Is it raining?	No	a question
Please sit down.	No	a command
$x + 2 = 5$	No	an open sentence: true for $x = 3$ , false otherwise (Lesson 4)
Every even number greater than 2 is the sum of two primes.	Yes	it is true or false, although nobody has yet proved which (Goldbach's conjecture)

The last row matters: a statement must have a truth value, not that we must know it.

George Boole (1815–1864). His book The Laws of Thought (1854) treated true and false like the numbers 1 and 0, with rules for combining them; this Boolean algebra is the foundation of digital circuit design and of modern computer science. Engraving, about 1865, public domain, via Wikimedia Commons.

Definition

Logical connectives. From statements $p$ and $q$ we build compound statements:

Negation $\neg p$ (also written $\sim p$ ), "not $p$ ": true exactly when $p$ is false.
Conjunction $p \wedge q$ , " $p$ and $q$ ": true only when both $p$ and $q$ are true.
Disjunction $p \vee q$ , " $p$ or $q$ ": true when at least one of $p, q$ is true (the inclusive or).
Exclusive disjunction $p \veebar q$ , "either $p$ or $q$ , but not both": true when exactly one is true.

Truth tables

A truth table lists every possible combination of truth values of the simple statements and the resulting value of the compound statement. With two statements there are $2 \times 2 = 4$ rows; with $n$ statements there are $2^{n}$ .

$p$	$q$	$\neg p$	$p \wedge q$	$p \vee q$	$p \veebar q$
T	T	F	T	T	F
T	F	F	F	T	T
F	T	T	F	T	T
F	F	T	F	F	F

In mathematics "or" always means the inclusive $\vee$ unless "but not both" is said. In English, "but", "although" and "yet" are all conjunctions: "It is raining but the match goes on" is $p \wedge q$ .

Worked example 1: truth values of compound statements

Let $p$ : "9 is a perfect square" and $q$ : "9 is even". Find the truth values of (a) $\neg q$ (b) $p \wedge q$ (c) $p \vee q$ (d) $\neg p \vee q$ (e) $\neg(p \wedge \neg q)$ .

Solution $p$ is T ( $9 = 3^{2}$ ) and $q$ is F.

$\neg q$ : T.
$p \wedge q$ : T and F, so F.
$p \vee q$ : T or F, so T.
$\neg p \vee q$ : F or F, so F.
$p \wedge \neg q$ is T and T, so T; its negation is F.

Worked example 2: from words to symbols

Let $p$ : "It is raining" and $q$ : "The match is cancelled". Write in symbols: (a) It is raining but the match is not cancelled. (b) It is neither raining nor is the match cancelled. (c) It is not true that it is raining and the match is cancelled.

Solution (a) $p \wedge \neg q$ (b) $\neg p \wedge \neg q$ (c) $\neg(p \wedge q)$ . Notice that (b) and (c) are different: (c) is true on a dry day when the match is cancelled, but (b) is false then.

Worked example 3: building a truth table

Construct the truth table of $(p \vee q) \wedge \neg(p \wedge q)$ and compare it with $p \veebar q$ .

Solution Work in columns, from the inside out.

$p$	$q$	$p \vee q$	$p \wedge q$	$\neg(p \wedge q)$	$(p \vee q) \wedge \neg(p \wedge q)$
T	T	T	T	F	F
T	F	T	F	T	T
F	T	T	F	T	T
F	F	F	F	T	F

The last column is F, T, T, F, exactly the column of $p \veebar q$ : " $p$ or $q$ , and not both" is the exclusive or.

Worked example 4: three statements

Construct the truth table of $p \wedge (q \vee \neg r)$ .

Solution Three statements give $2^{3} = 8$ rows.

$p$	$q$	$r$	$\neg r$	$q \vee \neg r$	$p \wedge (q \vee \neg r)$
T	T	T	F	T	T
T	T	F	T	T	T
T	F	T	F	F	F
T	F	F	T	T	T
F	T	T	F	T	F
F	T	F	T	T	F
F	F	T	F	F	F
F	F	F	T	T	F

The statement is true in exactly three rows: whenever $p$ is true, except when $q$ is false and $r$ is true.

Warning

The negation of "and" is not "not … and not …". $\neg(p \wedge q)$ and $\neg p \wedge \neg q$ have different truth tables (compare Worked example 2). Lesson 3 gives the correct rule, De Morgan's law: $\neg(p \wedge q)$ is equivalent to $\neg p \vee \neg q$ .

Tip

Fill the first columns of a truth table in a fixed pattern so that no row is missed or repeated: with $n$ statements, the first column is $2^{n-1}$ T's then $2^{n-1}$ F's, the next column halves the blocks, and the last column alternates T, F, T, F.

Dr. Trefor Bazett: Intro to Truth Tables, negation, conjunction and disjunction. Watch for how each column of a truth table is built from the columns before it.

✦

Practice

Use T and F for truth values.

Which are statements? Give the truth value of each statement. (a) 11 is a prime number. (b) Shut the window. (c) $x^{2} = 9$ (d) The angles of a triangle add up to $180^\circ$ . (e) Is 51 prime? (f) 51 is prime.
$p$ : "6 is even", $q$ : " $6 \gt 10$ ". Find the truth values of $\neg p$ , $p \wedge q$ , $p \vee q$ and $\neg p \vee \neg q$ .
With $p$ : "Aline studies" and $q$ : "Aline passes", write in symbols: (a) Aline studies and passes. (b) Aline studies but does not pass. (c) Aline neither studies nor passes. (d) Aline passes or does not study.
Construct the truth table of $\neg p \vee q$ .
Construct the truth tables of $\neg(p \vee q)$ and $\neg p \wedge \neg q$ . What do you notice?
How many rows has the truth table of a statement built from 4 simple statements?
Construct the truth table of $(p \wedge q) \vee \neg r$ .
$p \wedge q$ is false and $p \vee q$ is true. What can you say about $p$ and $q$ ?

Show answersHide answers

(a) Statement, T. (b) Not a statement. (c) Not a statement (open sentence). (d) Statement, T. (e) Not a statement. (f) Statement, F ( $51 = 3 \times 17$ ).
$p$ is T and $q$ is F: $\neg p$ F; $p \wedge q$ F; $p \vee q$ T; $\neg p \vee \neg q$ T.
(a) $p \wedge q$ (b) $p \wedge \neg q$ (c) $\neg p \wedge \neg q$ (d) $q \vee \neg p$ .
Rows TT, TF, FT, FF give T, F, T, T.
Both columns are F, F, F, T: the two statements always have the same truth value.
$2^{4} = 16$ .
Rows TTT to FFF give T, T, F, T, F, T, F, T.
Exactly one of $p$ and $q$ is true: $p \veebar q$ is true.

Quick check

Which of these is a statement?

Key points

A statement is a declarative sentence that is true or false, not both.
$\neg p$ reverses $p$ ; $p \wedge q$ needs both; $p \vee q$ needs at least one; $p \veebar q$ needs exactly one.
A truth table with $n$ simple statements has $2^{n}$ rows; build it column by column.

Sources and credits

Written by Ganzaa: the explanations, the tables, the worked examples, the practice items and the questions. The unit follows the topic list of REB's S4 Core Mathematics (2026–2027); no REB text is reproduced.
Boole's dates and work: George Boole, Wikipedia contributors, Wikipedia, CC BY-SA 4.0, adapted by Ganzaa.
Portrait: George Boole, engraving, about 1865, Wikimedia Commons, public domain.
Video: Intro to Truth Tables: Negation, Conjunction, and Disjunction, Dr. Trefor Bazett, embedded from YouTube.
This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1"And" is not the same as "or"

2Statement or not?

3Truth tables

"And" is not the same as "or"

Statement or not?

Truth tables