Logic

Q: Under what condition is a conjunction ($p \wedge q$) True?

A conjunction $p \wedge q$ is True only when both $p$ and $q$ are True; if either proposition is False, the conjunction is False.

Q: Under what condition is a disjunction ($p \vee q$) False?

A disjunction $p \vee q$ is False only when both $p$ and $q$ are False; it is True if at least one component is True.

Further Mathematics 2330 · CCEA GCSE · 10 min read · Updated: 31 Aug 2026

Welcome to Logic: Unit 4 (Discrete and Decision Mathematics)

Welcome to one of the most fascinating topics in Further Mathematics! Have you ever wondered how computers "think", how search engines filter results, or how lawyers construct bulletproof arguments? The answer is Logic.

In this chapter, you will learn how to turn everyday English statements into clear mathematical symbols, evaluate whether arguments are true or false, and build step-by-step truth tables. Don't worry if this feels like a completely new language at first — once you learn a few simple rules, working with logic is just like solving a fun puzzle!

Did you know? Modern computer processors are built from billions of microscopic switches called logic gates. The exact same rules of logic you are learning here are what power your smartphone and gaming consoles!

1. Statements and Propositions

In logic, we deal with special sentences called propositions (or statements).

A proposition is a declarative statement that is either strictly True ($\text{T}$) or strictly False ($\text{F}$), but never both at the same time.

We usually represent propositions using lower-case letters like $p$, $q$, and $r$.

Examples of Propositions:

• "Belfast is the capital of Northern Ireland." — This is a proposition with truth value True ($\text{T}$).
• "$5 + 3 = 12$" — This is a proposition with truth value False ($\text{F}$).
• "A triangle has three sides." — This is a proposition with truth value True ($\text{T}$).

What is NOT a Proposition?

• "What time is the bus?" — This is a question, so it cannot be true or false.
• "Pass the salt, please." — This is a command, not a factual statement.
• "Maths is the best subject." — This is an opinion, not an objective truth.

Key Takeaway: If you cannot definitively label a sentence as either True ($\text{T}$) or False ($\text{F}$), it is not a proposition!

2. The Core Logical Connectives

We can combine simple propositions to make compound statements using special words known as connectives (or operators). Let's look at the five main connectives you need for your CCEA exam.

1. Negation (NOT)

Symbol: $\sim p$ or $\neg p$
Meaning: "It is not the case that $p$" or simply "NOT $p$".
How it works: Negation simply flips the truth value of a proposition upside down. If $p$ is true, $\sim p$ is false. If $p$ is false, $\sim p$ is true.

Example: Let $p$ = "It is raining." Then $\sim p$ = "It is not raining."

Truth Table for Negation:
• When $p = \text{T}$, $\sim p = \text{F}$
• When $p = \text{F}$, $\sim p = \text{T}$

2. Conjunction (AND)

Symbol: $p \wedge q$
Meaning: "$p$ AND $q$"
Memory Aid: The symbol $\wedge$ looks like an uppercase letter A (for AND)!
How it works: For $p \wedge q$ to be true, both $p$ and $q$ must be true. If even one part is false, the whole statement is false.

Analogy: Imagine a teacher says: "To pass the test, you must bring a pen ($p$) AND bring a ruler ($q$)." If you only bring a pen, you have not followed the instruction.

Truth Table for Conjunction:
• $p = \text{T}, q = \text{T} \implies p \wedge q = \text{T}$
• $p = \text{T}, q = \text{F} \implies p \wedge q = \text{F}$
• $p = \text{F}, q = \text{T} \implies p \wedge q = \text{F}$
• $p = \text{F}, q = \text{F} \implies p \wedge q = \text{F}$

3. Disjunction (Inclusive OR)

Symbol: $p \vee q$
Meaning: "$p$ OR $q$" (or both)
How it works: A disjunction is true if at least one of the statements is true. It is only false when both statements are false.

Analogy: A cinema says: "You get a discount if you are a student ($p$) OR a senior citizen ($q$)." As long as you fit at least one category, you get the discount!

Truth Table for Disjunction:
• $p = \text{T}, q = \text{T} \implies p \vee q = \text{T}$
• $p = \text{T}, q = \text{F} \implies p \vee q = \text{T}$
• $p = \text{F}, q = \text{T} \implies p \vee q = \text{T}$
• $p = \text{F}, q = \text{F} \implies p \vee q = \text{F}$

4. Implication / Conditional (IF... THEN)

Symbol: $p \implies q$ or $p \rightarrow q$
Meaning: "If $p$, then $q$" (where $p$ is the hypothesis and $q$ is the conclusion).
How it works: An implication is broken (false) in only one specific case: when the starting condition $p$ is true, but the outcome $q$ turns out to be false.

Analogy (The Promise Test): A parent promises: "If you tidy your room ($p$), then I will give you \$10 ($q$)."
• Case 1 ($\text{T} \implies \text{T}$): You tidy your room, you get \$10. Promise kept! ($\text{T}$)
• Case 2 ($\text{T} \implies \text{F}$): You tidy your room, but you don't get \$10. Promise broken! ($\text{F}$)
• Case 3 ($\text{F} \implies \text{T}$): You don't tidy your room, but your parent is generous and gives you \$10 anyway. The original promise was never broken! ($\text{T}$)
• Case 4 ($\text{F} \implies \text{F}$): You don't tidy your room, and you don't get \$10. No promise was broken! ($\text{T}$)

Truth Table for Implication:
• $p = \text{T}, q = \text{T} \implies p \implies q = \text{T}$
• $p = \text{T}, q = \text{F} \implies p \implies q = \text{F}$
• $p = \text{F}, q = \text{T} \implies p \implies q = \text{T}$
• $p = \text{F}, q = \text{F} \implies p \implies q = \text{T}$

Common Mistake to Avoid: Many students mistakenly think that when $p$ is false, the implication must be false. Remember: if the starting condition $p$ is false ($\text{F}$), the implication $p \implies q$ is always automatically True ($\text{T}$)!

5. Equivalence / Biconditional (IF AND ONLY IF)

Symbol: $p \iff q$ or $p \leftrightarrow q$
Meaning: "$p$ if and only if $q$" (often abbreviated as "$p$ iff $q$").
How it works: This is true whenever $p$ and $q$ have the exact same truth value (both true or both false).

Truth Table for Equivalence:
• $p = \text{T}, q = \text{T} \implies p \iff q = \text{T}$
• $p = \text{T}, q = \text{F} \implies p \iff q = \text{F}$
• $p = \text{F}, q = \text{T} \implies p \iff q = \text{F}$
• $p = \text{F}, q = \text{F} \implies p \iff q = \text{T}$

Key Takeaway:
• AND ($\wedge$) requires BOTH to be true.
• OR ($\vee$) requires AT LEAST ONE to be true.
• IF... THEN ($\implies$) is ONLY false when $\text{T} \implies \text{F}$.
• EQUIVALENCE ($\iff$) is true when both values MATCH.

3. Constructing Truth Tables Step-by-Step

A truth table is a systematic way to list all possible combinations of truth values for individual propositions and calculate the final result of a compound expression.

Standard Starting Setup:

When working with two propositions, $p$ and $q$, there are always $2^2 = 4$ possible combinations. Always set up your first two columns in this standard order:

• Row 1: $p = \text{T}$, $q = \text{T}$
• Row 2: $p = \text{T}$, $q = \text{F}$
• Row 3: $p = \text{F}$, $q = \text{T}$
• Row 4: $p = \text{F}$, $q = \text{F}$

Worked Example 1: Construct a truth table for $\sim(p \wedge q)$

Step 1: Write the input columns for $p$ and $q$.
Step 2: Create an intermediate column for the brackets first: $(p \wedge q)$.
Step 3: Apply the negation $\sim$ to flip the values of the $(p \wedge q)$ column.

Result Row-by-Row:
• Row 1: $p = \text{T}, q = \text{T} \implies (p \wedge q) = \text{T} \implies \sim(p \wedge q) = \mathbf{F}$
• Row 2: $p = \text{T}, q = \text{F} \implies (p \wedge q) = \text{F} \implies \sim(p \wedge q) = \mathbf{T}$
• Row 3: $p = \text{F}, q = \text{T} \implies (p \wedge q) = \text{F} \implies \sim(p \wedge q) = \mathbf{T}$
• Row 4: $p = \text{F}, q = \text{F} \implies (p \wedge q) = \text{F} \implies \sim(p \wedge q) = \mathbf{T}$

The final column values from top to bottom are: $\text{F}, \text{T}, \text{T}, \text{T}$.

Worked Example 2: Construct a truth table for $(p \vee q) \wedge \sim p$

Let's build this step-by-step using separate columns:

• Row 1 ($p = \text{T}, q = \text{T}$):
$(p \vee q) = \text{T}$
$\sim p = \text{F}$
Final result: $\text{T} \wedge \text{F} = \mathbf{F}$

• Row 2 ($p = \text{T}, q = \text{F}$):
$(p \vee q) = \text{T}$
$\sim p = \text{F}$
Final result: $\text{T} \wedge \text{F} = \mathbf{F}$

• Row 3 ($p = \text{F}, q = \text{T}$):
$(p \vee q) = \text{T}$
$\sim p = \text{T}$
Final result: $\text{T} \wedge \text{T} = \mathbf{T}$

• Row 4 ($p = \text{F}, q = \text{F}$):
$(p \vee q) = \text{F}$
$\sim p = \text{T}$
Final result: $\text{F} \wedge \text{T} = \mathbf{F}$

The final column values from top to bottom are: $\text{F}, \text{F}, \text{T}, \text{F}$.

4. Logical Equivalence, Tautologies, and Contradictions

Logical Equivalence ($\equiv$)

Two compound statements are said to be logically equivalent if and only if they produce the exact same final truth values for every single row in a truth table.

Example: Show that $p \implies q$ is logically equivalent to $\sim p \vee q$.

Let's compare their final values row by row:
• Row 1 ($p = \text{T}, q = \text{T}$): $p \implies q = \mathbf{T}$ and $\sim p \vee q = \text{F} \vee \text{T} = \mathbf{T}$
• Row 2 ($p = \text{T}, q = \text{F}$): $p \implies q = \mathbf{F}$ and $\sim p \vee q = \text{F} \vee \text{F} = \mathbf{F}$
• Row 3 ($p = \text{F}, q = \text{T}$): $p \implies q = \mathbf{T}$ and $\sim p \vee q = \text{T} \vee \text{T} = \mathbf{T}$
• Row 4 ($p = \text{F}, q = \text{F}$): $p \implies q = \mathbf{T}$ and $\sim p \vee q = \text{T} \vee \text{F} = \mathbf{T}$

Because both columns give the exact sequence $\text{T}, \text{F}, \text{T}, \text{T}$, we write:
$(p \implies q) \equiv (\sim p \vee q)$

Tautology

A tautology is a statement that is always True, no matter what the truth values of the individual propositions are.

Example: The statement $p \vee \sim p$ ("It is raining OR it is not raining").
• If $p = \text{T}$, then $\text{T} \vee \text{F} = \text{T}$
• If $p = \text{F}$, then $\text{F} \vee \text{T} = \text{T}$
Since the output is always $\text{T}$, $p \vee \sim p$ is a tautology.

Contradiction (Fallacy)

A contradiction is a statement that is always False under every possible circumstance.

Example: The statement $p \wedge \sim p$ ("It is raining AND it is not raining").
• If $p = \text{T}$, then $\text{T} \wedge \text{F} = \text{F}$
• If $p = \text{F}$, then $\text{F} \wedge \text{T} = \text{F}$
Since the output is always $\text{F}$, $p \wedge \sim p$ is a contradiction.

5. Quick Summary & Exam Checklist

Before stepping into your Further Maths exam, make sure you can answer YES to all of these:

• Can I identify whether a sentence is a valid proposition?
• Do I know all 5 core symbols: NOT ($\sim$), AND ($\wedge$), OR ($\vee$), IF... THEN ($\implies$), and IFF ($\iff$)?
• Do I remember that $p \implies q$ is ONLY false when $p = \text{T}$ and $q = \text{F}$?
• Can I systematically set up a 4-row truth table with working columns?
• Can I prove logical equivalence by showing two columns have identical truth values?
• Do I know that a tautology is all $\text{T}$'s and a contradiction is all $\text{F}$'s?

Top Exam Tip: When building truth tables in your exam, always add intermediate columns for bracketed parts and negations. Breaking the problem down into simple individual steps is the easiest way to avoid simple mistakes and guarantee full marks!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is a proposition in mathematical logic?

A proposition (or statement) is a declarative sentence that is either strictly True ($\text{T}$) or strictly False ($\text{F}$), but never both at the same time.

What does the negation connective ($\sim p$ or $\neg p$) do to a proposition?

Negation flips the truth value of a proposition: if $p$ is True, $\sim p$ is False; if $p$ is False, $\sim p$ is True.

Under what condition is a conjunction ($p \wedge q$) True?

A conjunction $p \wedge q$ is True only when both $p$ and $q$ are True; if either proposition is False, the conjunction is False.

Under what condition is a disjunction ($p \vee q$) False?

A disjunction $p \vee q$ is False only when both $p$ and $q$ are False; it is True if at least one component is True.

When is an implication (conditional statement $p \implies q$) False?

An implication $p \implies q$ is False in only one specific case: when the hypothesis $p$ is True and the conclusion $q$ is False ($\text{T} \implies \text{F}$).

When is an equivalence (biconditional statement $p \iff q$) True?

An equivalence $p \iff q$ is True whenever $p$ and $q$ have the exact same truth value (both True or both False).

How many rows are needed in a standard truth table for two propositions $p$ and $q$, and in what order are the inputs listed?

A truth table with two propositions requires $2^2 = 4$ rows: $(\text{T}, \text{T})$, $(\text{T}, \text{F})$, $(\text{F}, \text{T})$, and $(\text{F}, \text{F})$.

How do you prove that two compound statements are logically equivalent ($\equiv$)?

Construct a truth table for both statements; they are logically equivalent if their final columns have identical truth values in every single row.

What is the difference between a tautology and a contradiction?

A tautology is a statement that is always True for all possible truth values, whereas a contradiction is a statement that is always False under every circumstance.

Ready to test yourself?

Turn these notes into exam-style practice. Get unlimited AI questions on this topic with instant marking and explanations.

Practise This Topic

More Further Mathematics 2330 chapters

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

Put These Notes into Practice

Reading the notes is a great start. Now practise with unlimited AI-generated questions and get instant feedback. 100,000+ students are already improving their grades.

Start Practising Now View Pricing

Done reading? Test yourself with AI practice questions

Practise This Topic Now