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: A conjunction P and Q is true exactly when both P and Q are true.

Common stuck point: The procedure for conjunction is the easy part; the trap is declaring $P \wedge Q$ true when only one part is true. Asking "Does the whole claim require every part to be true at the same time?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the whole claim require every part to be true at the same time?

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

Answer

p \land q = T,\quad p \land \neg q = F,\quad \neg p \land q = F

First step

p

is true (4 is even).

q

is true (4 < 10). So

\neg q

is false and

\neg p

is false.

Full solution

2
$p \land q = T \land T = T$ .
3
$p \land \neg q = T \land F = F$ .
4
$\neg p \land q = F \land T = F$ .

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

Example 3

medium

Use De Morgan's law to write

\neg(P \wedge Q)

in terms of

\neg P

and

\neg Q

Example 4

medium

Show the associative law:

(P \wedge Q) \wedge R \equiv P \wedge (Q \wedge R)

Example 5

medium

Negate the statement: 'Bob is tall AND Bob is fast.'

Example 6

hard

Build a truth table for

(P \wedge Q) \wedge \neg P

and identify it.

Example 7

medium

Show

P \wedge P \equiv P

(idempotence).

Example 8

hard

Determine if '(

P \vee Q

)

\wedge

(

P \vee \neg Q

)' simplifies to

P

Example 9

challenge

Find all integers

n

with

1 \le n \le 100

satisfying '

n

is divisible by

3

AND

n

is divisible by

5

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.

Example 3

easy

P

is true and

Q

is true, what is

P \wedge Q

Example 4

easy

P

is true and

Q

is false, what is

P \wedge Q

Example 5

easy

If both

P

and

Q

are false, what is

P \wedge Q

Example 6

easy

Evaluate: '

3>2

AND

5>4

Example 7

easy

Evaluate: '

2

is even AND

2

is prime'.

Example 8

easy

In everyday speech 'tall enough AND has a ticket', can you ride with only a ticket?

Example 9

easy

How many of the 4 truth-table rows make

P\wedge Q

true?

Example 10

easy

Evaluate: '

7

is even AND

7

is odd'.

Example 11

medium

For what

x

is '

x>0

AND

x<5

' true? Express as an interval.

Example 12

medium

Solve: '

x

is a multiple of 2 AND a multiple of 3', for positive integers. Describe the solutions.

Example 13

medium

P\wedge Q

is true, what can you conclude about

P

and about

Q

individually?

Example 14

medium

P \wedge Q

logically equivalent to

Q \wedge P

Example 15

medium

Translate to logic and evaluate: '

\sqrt{16}=4

and

0

is positive'.

Example 16

medium

For '

x^2=4

AND

x>0

', find all real

x

Example 17

medium

Build the truth table column for

P \wedge \neg Q

Example 18

medium

Evaluate '

P \wedge Q

' where

P

: '

10

is divisible by

5

' and

Q

: '

10

is divisible by

4

Example 19

medium

P\wedge Q

is false and

P

is true, what is

Q

Example 20

challenge

How many of the

2^3=8

rows make

P\wedge Q\wedge R

true, and why?

Example 21

challenge

Show '

P \wedge (Q \vee R)

' equals '

(P\wedge Q)\vee(P\wedge R)

' (distributivity).

Example 22

challenge

In a survey, a response counts only if 'consented AND completed AND age

\ge 18

'. Of 100 forms, 90 consented, 85 completed, 80 are adults. What is the minimum number of valid responses?

Example 23

easy

How many rows of the truth table for

P \wedge Q

are TRUE?

Example 24

easy

Evaluate: '

5

is odd AND

5

is prime'.

Example 25

easy

Evaluate: '

4

is even AND

4

is prime'.

Example 26

medium

Find all

x \in \mathbb{Z}

with '

x > -2

AND

x \le 3

Example 27

medium

Express the solution to '

x \ge 0

AND

x \le 5

' as an interval.

Example 28

medium

Determine the truth value of: '

\sqrt{2}

is irrational AND

2+2=4

Example 29

medium

True or false:

P \wedge (P \vee Q) \equiv P

Example 30

easy

Translate to symbols: 'It is raining AND it is cold.' Let

R

= raining,

C

= cold.

Example 31

medium

How many integers

n

satisfy '

1 \le n \le 10

AND

n

is prime'?

Example 32

medium

True or false:

P \wedge \neg P

is always false.

Example 33

medium

Solve: '

x^2 = 9

AND

x > 0

Example 34

medium

Solve over

\mathbb{R}

: '

x>0

AND

\sin x = 0

AND

x < 2\pi

Example 35

medium

P

is true and

\neg Q

is true, what is

P \wedge Q

Example 36

medium

Translate the inequality

1 \le x \le 5

as a conjunction.

Example 37

hard

How many of the

2^3=8

rows of the truth table for

P \wedge Q \wedge R

are true?

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Related Concepts

Logical Statement Disjunction Truth Table

Background Knowledge

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

logical statement