# Discrete Math Predicate Logic with Balls

Attempting to use the predicate symbols shown and appropriate quantifiers, write each English language statement as a predicate wff. (The domain is the whole world.) I want to know if this is correct.

B(x) is “x is a ball.”

R(x) is “x is round.”

S(x) is “x is a soccer ball.”

1. All balls are round.

My attempt: (∀x)B(x)∧[R(x)]

1. Not all balls are soccer balls.

My attempt: ~[(∀x)(B(x)→S(x))]

1. All soccer balls are round.

My attempt: (∀x)S(x)→[R(x)]

1. Some balls are not round.

My Attempt: (∃x)(B(x)∧~[R(x)]

(1) Incorrect. You've stated that anything in the whole world is round and is a ball. Answer: $$\forall x, \, B(x) \implies R(x).$$
(2) Correct. Simplification will give $$\exists x : B(x) \wedge \neg S(x).$$