Logical Statement Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

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

Solution

  1. 1
    The original statement has the form āˆ€xāˆˆP,ā€…ā€ŠxĀ isĀ odd\forall x \in P,\; x \text{ is odd} where PP is the set of primes.
  2. 2
    The negation is āˆƒxāˆˆP\exists x \in P such that xx is not odd, i.e., 'There exists a prime number that is not odd.'
  3. 3
    In fact, 22 is a prime number that is even, so the negation is true and the original statement is false.

Answer

ThereĀ existsĀ aĀ primeĀ numberĀ thatĀ isĀ even.\text{There exists a prime number that is even.}
To negate a universal statement āˆ€x,P(x)\forall x, P(x), form the existential āˆƒx,Ā¬P(x)\exists x, \neg P(x). A single counterexample suffices to disprove a universal claim.

About Logical Statement

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).

Learn more about Logical Statement ā†’

More Logical Statement Examples

Example 1 easy

Classify each as a statement (proposition) or not: (a) [formula] (b) 'Close the door.' (c) [formula]

Example 3 easy

Determine the truth value of each: (a) [formula] is a prime number. (b) [formula] is a natural numbe

Example 4 easy

Which of the following are logical statements: (a) '[formula] is prime', (b) 'Open the window', (c)

Example 5 medium

Classify each as a statement or not: (a) '[formula]' (b) 'Is 5 prime?' (c) 'Every square is a rectan

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