# Mathematical Statements and Logic

Mathematics I · Mathematical Language and Reasoning · https://tryals.app/learn/mathematics-i/mathematical-statements-and-logic

## Mathematical Statements

Mathematics relies on a language of absolute precision. Its core building blocks are **statements** (propositions): declarative assertions that are unambiguously either **true** or **false**, never both and never neither.

| Type | Example | Status |
| --- | --- | --- |
| Statement | "$7$ is prime" | True |
| Statement | "$4 > 9$" | False |
| Non-statement | "$x + 1$" | Neither |
| Non-statement | "Is this prime?" | Neither |

Simple statements combine into compound ones using **logical connectives**. **Negation** ($\neg p$) reverses truth value. **Conjunction** ($p \land q$) is true only when **both** inputs are true. **Disjunction** ($p \lor q$) is true when **at least one** input is true.

The meaning of any connective is captured by a **truth table** listing every combination of inputs. A compound of $n$ statements requires $2^n$ rows.

## Connectives and Equivalences

Two statements are **logically equivalent** when they share identical truth tables. This is the bedrock of proof: substituting equivalent statements preserves truth.

| Connective | Symbol | Truth Condition |
| --- | --- | --- |
| Negation | $\neg p$ | True when $p$ is false |
| Conjunction | $p \land q$ | True when both are true |
| Disjunction | $p \lor q$ | True when at least one is true |

$$p \lor q \text{ is true} \iff p \text{ or } q \text{ (or both)}$$

**Common Pitfall:** Never read mathematical **or** as the everyday **exclusive** or. In math, $p \lor q$ is **inclusive**: it is true when $p$ holds, when $q$ holds, or when both hold. For example, "$x > 0$ or $x < 5$" holds for $x = 3$ because both parts are true.

## Practice questions

4 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. A proposition must have a definite truth value. Sort each sentence.

**Answer:**

- Proposition: 17 is prime, Every even number above 2 is composite, 3 > 8
- Not a proposition: x + 1, Is 12 divisible by 5?, This sentence is false

**Why:** '17 is prime' (true), 'every even number above 2 is composite' (true) and '3 > 8' (false) all assert something definite. 'x + 1' names a number instead of claiming anything, a question asserts nothing, and 'this sentence is false' can be neither true nor false, the liar paradox.

Page: https://tryals.app/practice/mathematics-i/mathematical-statements-and-logic/a-proposition-must-have-a-definite-truth-value-sort-each-sentence

### 2. "x + 1" and "Is 7 prime?" are both rejected as mathematical statements, for what looks like the same reason. Why does mathematics insist on a definite truth value, when ordinary language manages without one?

A. Because logic operates on truth values, not on meanings.
B. Because mathematics only concerns itself with numbers, and questions are not numerical objects.
C. Because ordinary language is imprecise, whereas mathematical language is designed to avoid ambiguity of any kind.
D. Because both expressions contain a variable or a query, and mathematical statements may contain neither.

**Answer:** A. Because logic operates on truth values, not on meanings.

**Why:** The whole apparatus runs on truth values — 'and' is a table over true and false, modus ponens is a rule about preserving truth — so an expression with none gives the machinery nothing to compute with. The negation of 'Is 7 prime?' is not a false statement; it is not a statement. Variables are welcome: quantify one and you get a statement directly.

Page: https://tryals.app/practice/mathematics-i/mathematical-statements-and-logic/x-1-and-is-7-prime-are-both-rejected-as-mathematical

### 3. "This sentence is false." Why does the liar paradox matter to a subject that insists every statement is either true or false?

A. Because it proves that logic is inconsistent
B. Because it is grammatically malformed and therefore not really a sentence
C. Because it is a well-formed sentence that can be neither
D. Because it shows some statements have two truth values at once

**Answer:** C. Because it is a well-formed sentence that can be neither

**Why:** Suppose it true and it is false; suppose it false and it is true. Nothing is malformed, which is why mathematics restricts itself to a language where such self-reference cannot be built. Russell's paradox is the same shape in set theory, and it is why the axioms are written as carefully as they are.

Page: https://tryals.app/practice/mathematics-i/mathematical-statements-and-logic/this-sentence-is-false-why-does-the-liar-paradox-matter-to-a

### 4. Let p be TRUE and q be TRUE. Select every statement that is FALSE.

A. p OR (NOT q)
B. (NOT p) AND q
C. p AND q
D. NOT (p OR q)
E. NOT p

**Answer:** B. (NOT p) AND q; D. NOT (p OR q); E. NOT p

**Why:** With both true: p AND q is true; NOT p is false; p OR (NOT q) is true because p holds; (NOT p) AND q is false because NOT p fails; NOT (p OR q) is NOT true = false. So the false ones are options 2, 4 and 5.

Page: https://tryals.app/practice/mathematics-i/mathematical-statements-and-logic/let-p-be-true-and-q-be-true-select-every-statement-that-is-false