Question

Let K(x): x is a two-wheeler, L(x) : x is a scooter, M(x) : x is manufactured by Bajaj. Express the following using quantifiers.

I. Every two wheeler is a scooter.
II. There is a two-wheeler that is not manufactured by Bajaj.
III. There is a two-wheeler manufactured by Bajaj that is not a scooter.
IV. Every two-wheeler that is a scooter is manufactured by Bajaj.​

Answer 1

I. Every two-wheeler is a scooter. II. There is a two-wheeler that is not manufactured by Bajaj. III. There is a two-wheeler manufactured by Bajaj that is not a scooter. IV. Every two-wheeler that is a scooter is manufactured by Bajaj.

I. Every two wheeler is a scooter can be expressed as ∀x(K(x) → L(x)). This translates to 'For every x, if x is a two-wheeler, then x is a scooter.'

II. There is a two-wheeler that is not manufactured by Bajaj can be expressed as ∃x(K(x) ∧ ¬M(x)). This translates to 'There exists an x, such that x is a two-wheeler and x is not manufactured by Bajaj.'

III. There is a two-wheeler manufactured by Bajaj that is not a scooter can be expressed as ∃x(K(x) ∧ M(x) ∧ ¬L(x)). This translates to 'There exists an x, such that x is a two-wheeler, x is manufactured by Bajaj, and x is not a scooter.'

IV. Every two-wheeler that is a scooter is manufactured by Bajaj can be expressed as ∀x(K(x) ∧ L(x) → M(x)). This translates to 'For every x, if x is a two-wheeler and x is a scooter, then x is manufactured by Bajaj.'