Mathematical reasoning begins with the idea of a statement (also called a proposition). A sentence is a mathematically acceptable statement if it is either true or false — but not both at the same time. Every statement carries exactly one definite truth value.
This rules out a surprising amount of ordinary language. Questions ("Where do you live?"), commands ("Open the door"), exclamations ("How beautiful!") and opinions ("Mathematics is the most interesting subject") are not statements, because none of them can be assigned a single, fixed truth value. Sentences whose truth depends on an unspecified variable — "$x$ is greater than $5$" — are also not statements until the variable is pinned down, since the same sentence is true for some $x$ and false for others.
| Sentence | Statement? | Reason |
|---|---|---|
| $8$ is less than $6$. | Yes (false) | has a definite truth value |
| The sum of $2$ and $3$ is $5$. | Yes (true) | has a definite truth value |
| Close the window. | No | a command |
| What a hot day! | No | an exclamation |
| $x$ is an even number. | No | truth depends on $x$ |
| Tomorrow is Friday. | No | "tomorrow" is ambiguous |
Negation. The negation of a statement $p$ denies it; we write it $\sim p$ (read "not $p$"). If $p$ is true then $\sim p$ is false, and vice versa — they always carry opposite truth values. In words you usually form the negation by inserting "not", or by writing "It is not the case that …" / "It is false that …". For example, the negation of "$\sqrt{2}$ is a rational number" is "$\sqrt{2}$ is not a rational number".
| $p$ | $\sim p$ |
|---|---|
| T | F |
| F | T |
Simple and compound statements. A simple statement cannot be broken into two or more statements. A compound statement is formed by joining simple statements (called its component statements) with words such as "and", "or", "if-then", "if and only if". These joining words are the connectives. To analyse a compound statement you identify its components and the connective linking them.
The connective "And" ($\wedge$). The compound statement $p \wedge q$ ("$p$ and $q$") is true only when both $p$ and $q$ are true; if either component is false, the whole statement is false. Watch a subtlety: "and" is a logical connective only when it joins two statements. In "$2$ and $4$ are even numbers", the word "and" is part of the description of a single fact and the sentence is not a compound of two separate statements in the connective sense — whereas "Ravi is tall and Ravi is fair" genuinely joins two component statements.
| $p$ | $q$ | $p \wedge q$ | $p \vee q$ |
|---|---|---|---|
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | F |
The connective "Or" ($\vee$). The compound statement $p \vee q$ ("$p$ or $q$") is false only when both components are false; otherwise it is true. The mathematical "or" is, by default, the inclusive or: $p \vee q$ allows the possibility that $p$ and $q$ are both true. "A student who has taken Mathematics or Computer Science can apply" means a student with one subject, the other, or both may apply.
Everyday English sometimes uses the exclusive or, where exactly one of the two — but not both — can hold: "Two lines intersect at a point or they are parallel" cannot have both true at once. You decide which "or" is meant from the context. Unless stated otherwise in this chapter, "or" is inclusive.
Deeper Insight — why a "definite truth value" is the whole foundation: The single requirement that a statement be true or false (never both, never neither) is what makes mathematics provable rather than merely persuasive. Once each component has a fixed truth value, a connective is just a rule for combining those values — which is exactly what a truth table records. So the entire chapter is built on two moves: first decide whether a sentence even qualifies as a statement, and only then combine it. The most common slip is treating an opinion or an open sentence with a variable as if it were a statement; guard against that and negations, "and"/"or", and the implications that follow all behave predictably. Notice too how closely "$\sim$", "$\wedge$" and "$\vee$" echo the set operations complement, intersection and union — that parallel is not a coincidence, and recognising it lets you reuse De Morgan's laws in both worlds.