Disjunction Examples in Math

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

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 disjunction $P \vee Q$ is a compound statement that is true whenever at least one of its parts is true.

At least one must be true. Logical OR is inclusive — "P or Q or both" — unlike the exclusive everyday "either/or."

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 disjunction P or Q is false only when both parts are false.

Common stuck point: The procedure for disjunction is the easy part; the trap is reading logical 'or' as exclusive. Asking "Is the claim true as soon as at least one part is true?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the claim true as soon as at least one part is true?

Worked Examples

Example 1

easy

Let

p

: '

2

is even' and

q

: '

2

is odd'. Evaluate

p \lor q

and explain the result.

Answer

p \lor q = \text{True}

First step

p

: '

2

is even' — True.

q

: '

2

is odd' — False.

Full solution

2
$p \lor q = T \lor F = T$ .
3
In logic, $p \lor q$ (inclusive or) is true when at least one of $p$ , $q$ is true.

The logical OR is inclusive: it is true whenever at least one operand is true, and false only when both are false. This differs from the everyday exclusive 'or' which excludes the case where both are true.

Example 2

medium

Find all real

x

satisfying '

x < -1

x > 3

', and express as a union of intervals.

Example 3

easy

Evaluate '

\pi > 3

\pi < 1

' and identify which disjunct made it true.

Example 4

medium

Show by truth table that

P \vee (Q \vee R) \equiv (P \vee Q) \vee R

(associativity).

Example 5

medium

Evaluate: '

\sin(0)=0

\cos(0)=0

'. Identify the true disjunct.

Example 6

hard

Prove

(P \Rightarrow Q) \equiv (\neg P \vee Q)

via truth table.

Example 7

challenge

Prove that

\{\vee, \neg\}

is functionally complete: every truth function can be expressed using only OR and NOT.

Practice Problems

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

Example 1

easy

Evaluate: (a) 'F

\lor

T', (b) 'F

\lor

F', (c) 'T

\lor

T'.

Example 2

medium

Use a truth table to verify De Morgan's Law:

\neg(p \lor q) \equiv \neg p \land \neg q

Example 3

easy

P

is true and

Q

is false, what is

P \vee Q

Example 4

easy

If both

P

and

Q

are true, what is

P \vee Q

(inclusive or)?

Example 5

easy

If both

P

and

Q

are false, what is

P \vee Q

Example 6

easy

Evaluate: '

3>5

2<4

Example 7

easy

Evaluate: '

6

is prime OR

6

is even'.

Example 8

easy

How many of the 4 truth-table rows make

P\vee Q

true?

Example 9

easy

Does logical OR mean 'exactly one'?

Example 10

easy

Evaluate: '

1=2

3=3

Example 11

medium

For what

x

is '

x<0

x>5

' true? Express with intervals.

Example 12

medium

Solve: '

x

is a multiple of 2 OR a multiple of 3' for

1

12

. How many such

x

Example 13

medium

P\vee Q

is true and

P

is false, what must

Q

be?

Example 14

medium

P \vee Q

logically equivalent to

Q \vee P

Example 15

medium

Evaluate: '

\sqrt{9}=3

2+2=5

Example 16

medium

Solve '

x^2 - 5x + 6 = 0

' and explain why the answer is naturally an OR.

Example 17

medium

Build the truth table column for

\neg P \vee Q

and identify the familiar connective it equals.

Example 18

medium

Evaluate '

P \vee Q

' where

P

: '

7

is even' and

Q

: '

7

is prime'.

Example 19

medium

P\vee Q

is false, what are

P

and

Q

Example 20

challenge

How many of the

2^3=8

rows make

P\vee Q\vee R

true, and why?

Example 21

challenge

Show

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

(distributivity of OR over AND).

Example 22

challenge

A password is valid if it 'contains a digit OR is at least 12 characters'. Of 50 passwords, 30 contain a digit, 25 are long, 12 are both. How many are valid?

Example 23

easy

P

is false and

Q

is true, what is

P \vee Q

Example 24

easy

Evaluate: '

5

is even OR

5

is odd'.

Example 25

easy

How many of the

2^2=4

truth-table rows for

P \vee Q

are false?

Example 26

easy

True or false:

P \vee P \equiv P

Example 27

medium

Solve for

x

|x - 2| > 3

. Express the answer as a disjunction of inequalities.

Example 28

medium

Solve

x^2 - x - 6 = 0

and write the solution as a disjunction.

Example 29

medium

A number

n

from

1

20

satisfies '

n

is prime OR

n

is a perfect square'. How many such

n

Example 30

medium

P \vee Q

is true and

Q

is false, what must

P

be?

Example 31

medium

Show that

P \vee \text{T} \equiv \text{T}

for any

P

(domination law).

Example 32

medium

Translate to symbols: 'It is raining or it is snowing'. Use

R

for raining,

S

for snowing.

Example 33

medium

Find all integers

n

with

1 \le n \le 20

such that '

n

is divisible by

5

n

is divisible by

7

Example 34

hard

Out of 80 students, 50 take Spanish, 35 take French, and 20 take both. How many take Spanish OR French?

Example 35

hard

Express XOR (

P \oplus Q

) using only

\vee

\wedge

, and

\neg

Example 36

hard

Among the 16 possible truth functions of two variables, how many are true on at least one of the four rows? (Disjunction is one of them.)

Example 37

hard

A switch is 'on' if either of two sensors triggers. Sensor A triggers with probability

0.4

, Sensor B with

0.5

; they are independent. Probability the switch is on?

Example 38

challenge

In a group of 100 people, 60 like coffee, 50 like tea, 30 like juice; 20 like coffee and tea, 15 like coffee and juice, 10 like tea and juice, and 5 like all three. How many like at least one beverage?

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Related Concepts

Logical Statement Conjunction Truth Table

Background Knowledge

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

logical statement