Question

Express each of these statements using quantifiers. Next, express the negation in simple English. (Do not simply use the phrase "It is not the case that.")

a) Some old cats can learn new tricks.
b) Every fish can swim

Answer 1

The statements are expressed using quantifiers and their negations simplified in English: 'Some old cats can learn new tricks' as 'There are no old cats that can learn new tricks', and 'Every fish can swim' as 'There is at least one fish that cannot swim'.

Step-by-step explanation:

Let's begin by expressing the statements using quantifiers:

• a) Some old cats can learn new tricks.
  This can be expressed as ∃x(O(x) ∧ C(x) → L(x)), where ∃ stands for 'there exists', O(x) means 'x is an old cat', C(x) means 'x is a cat', and L(x) means 'x can learn new tricks'.
• b) Every fish can swim.
  This can be expressed as ∀x(F(x) → S(x)), where ∀ stands for 'for all', F(x) means 'x is a fish', and S(x) means 'x can swim'.

The negations in simple English:

• a) There are no old cats that can learn new tricks.
• b) There is at least one fish that cannot swim.