Practice Logical Statement in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy
Is 'Pizza is delicious' a logical statement?

Example 2

medium
Which of these has a truth value: (a) '2\sqrt{2} is irrational', (b) 'Hello!', (c) '3>13 > 1'?

Example 3

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

Example 4

challenge
Is 'P→PP \to P' a logical statement for any statement PP, and what is its truth value?

Example 5

easy
Is the sentence 'Hello!' a logical statement?

Example 6

easy
Is '3<53 < 5 and 5<75 < 7' a logical statement?

Example 7

easy
Is '2+2=52 + 2 = 5' a logical statement?

Example 8

medium
Determine the truth value: 'There exists a real xx with x2=−1x^2=-1.'

Example 9

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

Example 10

medium
Classify: (a) 'The set {1,2,3}\{1,2,3\} has three elements.' (b) '2\sqrt{2} is rational.' (c) 'Solve x+1=4x+1=4.'

Example 11

challenge
Given statements PP and QQ, explain why exactly one of 'PP' and '¬P\neg P' is true, and what this principle is called.

Example 12

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

Example 13

medium
Classify each as a statement or not: (a) '7+3=117 + 3 = 11' (b) 'Is 5 prime?' (c) 'Every square is a rectangle.'

Example 14

easy
Is 'All squares are rectangles' a logical statement?

Example 15

easy
Is 'The Earth is flat' a logical statement?

Example 16

easy
Is '5+5=115+5=11' a logical statement?

Example 17

hard
Negate '∀x∃y:x+y=0\forall x \exists y: x+y=0.'

Example 18

hard
Negate 'For every integer nn, if nn is prime then nn is odd.'

Example 19

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

Example 20

medium
Write the contrapositive of 'If nn is divisible by 44, then nn is even.'
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