Conjunction Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Conjunction.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A conjunction P \wedge Q is a compound statement that is true if and only if both constituent statements P and Q are individually true.

To enter a theme park ride, you must be tall enough AND have a valid ticket—both conditions must hold. If you are tall enough but lost your ticket, you cannot ride. A conjunction P \wedge Q works the same way: it is true only when every single part is true, and false the moment any part fails.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: P \wedge Q is true only when both P and Q are individually true; one false part makes the whole conjunction false.

Common stuck point: In everyday language, 'and' sometimes means 'or'—logic is stricter.

Sense of Study hint: Write out the truth table: fill in all four rows (TT, TF, FT, FF) and confirm only the TT row gives T.

Worked Examples

Example 1

easy
Let p: '4 is even' and q: '4 < 10'. Evaluate p \land q, p \land \neg q, and \neg p \land q.

Solution

  1. 1
    p is true (4 is even). q is true (4 < 10). So \neg q is false and \neg p is false.
  2. 2
    p \land q = T \land T = T.
  3. 3
    p \land \neg q = T \land F = F.
  4. 4
    \neg p \land q = F \land T = F.

Answer

p \land q = T,\quad p \land \neg q = F,\quad \neg p \land q = F
A conjunction p \land q is true only when both components are true. If either component is false, the conjunction is false.

Example 2

medium
Construct the full truth table for p \land q and use it to show that conjunction is commutative: p \land q \equiv q \land p.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Determine the truth value of: (a) '3 > 2 and 3 < 5', (b) '3 > 2 and 3 > 5'.

Example 2

medium
Simplify: find all x \in \mathbb{R} satisfying 'x > 1 and x < 4', and express as an interval.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

logical statement