Logical Statement Examples in Math

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

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 logical statement (or proposition) is a declarative sentence that has exactly one truth value: it is either true or false. For example, '7 is prime' is a logical statement (true), while 'Is 7 prime?' is not (it's a question).

A logical statement is any claim that can be judged definitively as true or false — questions, commands, and paradoxes are not statements.

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 logical statement is a declarative sentence with one definite truth value, true or false.

Common stuck point: The procedure for logical statement is the easy part; the trap is treating a question or command as a statement. Asking "Can this sentence, in principle, be labeled either true or false (and only one)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can this sentence, in principle, be labeled either true or false (and only one)?

Worked Examples

Example 1

easy

Classify each as a statement (proposition) or not: (a)

3 + 5 = 8

(b) 'Close the door.' (c)

x > 2

Answer

\text{(a) Statement (true), (b) Not a statement, (c) Not a statement (open sentence)}

First step

(a)

3 + 5 = 8

is a declarative sentence that is true. It is a statement.

Full solution

2
(b) 'Close the door' is a command, not declarative. It is not a statement.
3
(c) $x > 2$ has an unspecified variable, so its truth value is undetermined. It is an open sentence, not a statement.

A logical statement must be a declarative sentence with a definite truth value—either true or false, not both and not indeterminate.

Example 2

medium

Negate the statement: 'All prime numbers are odd.'

Example 3

medium

Classify each as a statement or not: (a) '

7 + 3 = 11

' (b) 'Is 5 prime?' (c) 'Every square is a rectangle.'

Example 4

medium

Classify: (a) 'The set

\{1,2,3\}

has three elements.' (b) '

\sqrt{2}

is rational.' (c) 'Solve

x+1=4

Example 5

medium

Negate 'Some students wear hats.'

Example 6

medium

Find the truth value of '

5

is prime OR

4

is prime.'

Example 7

medium

Find the truth value of '

3<5

AND

7<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 each: (a)

7

is a prime number. (b)

0

is a natural number. (c) Every square is a rectangle.

Example 2

easy

Which of the following are logical statements: (a) '

7

is prime', (b) 'Open the window', (c) 'What is

2+5

?', (d) '

3^2 = 10

Example 3

easy

Is '7 is prime' a logical statement?

Example 4

easy

Is 'Close the door' a logical statement?

Example 5

easy

Is 'Is 7 prime?' a logical statement?

Example 6

easy

Is 'The Earth is flat' a logical statement?

Example 7

easy

Is 'Pizza is delicious' a logical statement?

Example 8

easy

Is '

2 + 2 = 5

' a logical statement?

Example 9

easy

Is '

x + 1 = 3

' (with

x

unspecified) a logical statement?

Example 10

easy

Is 'All squares are rectangles' a logical statement?

Example 11

medium

Classify each: (a) 'It is raining', (b) 'What time is it?', (c) 'Do your homework'. Which are logical statements?

Example 12

medium

Is the sentence 'This statement is false' a logical statement?

Example 13

medium

Negate the statement 'All cats are black'. Is the negation a logical statement?

Example 14

medium

Is '

n

is even' a statement when

n

is a specific given integer, say

n = 6

Example 15

medium

Which of these has a truth value: (a) '

\sqrt{2}

is irrational', (b) 'Hello!', (c) '

3 > 1

Example 16

medium

Combine two statements

P

: '5 is odd' and

Q

: '5 is prime' with 'and'. Is '

P

and

Q

' a logical statement, and what is its truth value?

Example 17

medium

Is 'There exists an integer

x

with

x^2 = 4

' a logical statement?

Example 18

challenge

Explain why a sentence must have EXACTLY ONE truth value to be a logical statement, using 'This sentence is false' as a counterexample.

Example 19

challenge

Is '

P \to P

' a logical statement for any statement

P

, and what is its truth value?

Example 20

challenge

Given statements

P

and

Q

, explain why exactly one of '

P

' and '

\neg P

' is true, and what this principle is called.

Example 21

medium

Is 'Welcome to math class' a logical statement?

Example 22

medium

Is '

5 < 3

' a logical statement, and what is its truth value?

Example 23

easy

Is '

5+5=11

' a logical statement?

Example 24

easy

True or false: every logical statement is either true or false.

Example 25

medium

Negate the statement '

x>3

for all real

x

Example 26

medium

Write the negation of '

n

is even'.

Example 27

easy

Is '

x+3

' a logical statement?

Example 28

medium

Determine the truth value: 'There exists a real

x

with

x^2=-1

Example 29

medium

Which of these are open sentences (not statements until

x

is bound)? (a)

x^2=4

(b)

\forall x, x^2\ge 0

(c)

x

is prime.

Example 30

hard

Negate 'For every integer

n

, if

n

is prime then

n

is odd.'

Example 31

easy

Is '

3 < 5

and

5 < 7

' a logical statement?

Example 32

medium

Write the contrapositive of 'If

n

is divisible by

4

, then

n

is even.'

Example 33

easy

Is the sentence 'Hello!' a logical statement?

Example 34

medium

State the converse of 'If it is raining, the ground is wet.'

Example 35

hard

Is the statement 'If

0=1

, then

2=3

' true or false?

Example 36

hard

Negate '

\forall x \exists y: x+y=0

Example 37

medium

Is '

2+2=4

if and only if the sky is blue' a logical statement?

Example 38

challenge

Liar paradox: 'This sentence is false.' Is it a logical statement?

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Related Concepts

Negation Conjunction Disjunction