Answer
a. False. All the triangles are gray. None of the circles are gray. Hence there are no triangles and circles that have the same color.
b. Formal version: ∃x(Circle(x) ∧ (∃y(Triangle(y) ∧ SameColor(x, y))))
c. Formal negation: ∀x(∼Circle(x) ∨ (∀y(∼Triangle(y) ∨ ∼SameColor(x, y))))
Work Step by Step
c. ~(∃x(Circle(x) ∧ (∃y(Triangle(y) ∧ SameColor(x, y)))))
≡ ∀x ~(Circle(x) ∧ (∃y(Triangle(y) ∧ SameColor(x, y))))
(by the law of negating an ∃ statement)
≡ ∀x (~Circle(x) ∨ ~(∃y(Triangle(y) ∧ SameColor(x, y))))
(by De Morgan's Law)
≡ ∀x (~Circle(x) ∨ (∀y ~(Triangle(y) ∧ SameColor(x, y))))
(by the law of negating an ∃ statement)
≡ ∀x (~Circle(x) ∨ (∀y (~Triangle(y) ∨ ~SameColor(x, y))))
(by De Morgan's law)
