Express the quantified logical statements using the predicates

Question

I just started learning predicates and quantifiers. I am pretty confused so I was wondering if someone can help me.

Using the predicates $P(x)$ to denote “x is a pro baseball player”, $R(x)$ to denote “x is rich”, $L(x)$ to denote “x is a pro football player” and $K(x, y)$ to denote “x knows y”, write down quantified logical statements toexpress:

1. All pro football player are rich.
2. Some pro baseball players are rich.
3. All pro baseball players know at least one pro football player.
4. All pro baseball players know a rich a pro football player.
5. Some pro football players know a rich pro baseball player.
6. Everyone knows a rich pro football player or a rich pro baseball player.

The domain of discourse is all people in the world.

This is what I have so far:

1. $∀x (L(x) ⇒ R(x))$
2. $∃x (P(x) ∧ R(x))$
3. $∀x(P(x)⇒∃y(K(x,y)∧L(y)))$

Unsure about 4 and 5

4. $∀x∃y(P(x)⇒K(x,y)∧(R(y)∧(L(y)))$
5. $∃x∃y(L(x)∧(K(x,y)∧(R(y)∧(P(y)))$
6. $∀x∃y(K(x,y)∧R(y)∧(P(y)∨L(y)))$

Answer 1

(1) is correct. "If anyone is a pro football player then they are rich" $$\forall x\big(L(x)\to R(x)\big)$$

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For (2), a restriction on an existential quantifier does not use the implication, but the conjunction.

"Someone is a pro baseball player and is rich" $$\exists x\big(P(x) \wedge R(x)\big)$$

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For (3) you can't use predicates as arguments for the predicate $K$.

3: All pro baseball players know at least one pro football player.

"If anyone $x$ is a pro baseball player, then there's someone $y$ that is known by $x$ and is a pro football player"

$$\forall x \Big(P(x) \to \exists y \big(K(x,y)\wedge L(y)\big)\Big)$$

Or equivalently $\;\forall x \exists y \Big(P(x) \to \big(K(x,y)\wedge L(y)\big)\Big)\;$ if you must move quantifiers to the left.

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Can you do the rest now?

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This is what I have so far:

All pro baseball players know a rich a pro football player.

$∀x∀y(P(x)⇒K(x,y)∧L(y))$

That says "if anyone is a pro baseball player then everyone is known by them and a pro football player"

You want to say: "If anyone is a pro baseball player then they know someone who is rich and a pro football player"

Some pro football players know a rich pro baseball player.

$∃x(P(x)⇒∀y(K(x,y)∧L(y)))$

W...? That says: "There is someone who is either not a pro baseball player or else knows everyone and everyone is a pro football player."

What you want to say is "Someone is a pro football player and knows someone who is rich and a pro baseball player."

Everyone knows a rich pro football player or a rich pro baseball player.

$∀x(∃y(K(x,R(P(y))∨K(x,R(L(y)))$

... Don't try to mash the predicates together. Use conjunctions. But you do seem to be on the right trail.

What seem to be trying to say is "everyone knows someone who is rich and either a pro football player or a pro baseball player." Thus:

$$\forall x \exists y\Big( K(x,y) \wedge R(y)\wedge \big(P(y)\vee L(y)\big) \Big)$$