Help with simple logic questions

[XodoX]

1. Mark as true or false. The implication "If P, then Q " is equivalent to:

a) P is necessary for Q
b) Q is sufficient for P
c) P is sufficient for Q
d) Q is necessary for P
e) P only if Q
f) Q only if P

It seems like there's missing something..If P is what? I don't get this.



2. Using predicates S(x) for "x is smart" , M(x) for "x makes a lot of money" , formalize the following satements.

a) Some smart people make a lot of money.
b) Not all smart people make a lot of money.
c) People who are smart make a lot of money
d) Only smart people make a lot of money.
e) There are people who are not smart but make a lot of money.


I don't get this one either. How am I supposed to use S(x) and M(x) here?


Thank you!

 

[Elucidus]

1. Mark as true or false. The implication "If P, then Q " is equivalent to:

a) P is necessary for Q
b) Q is sufficient for P
c) P is sufficient for Q
d) Q is necessary for P
e) P only if Q
f) Q only if P

It seems like there's missing something..If P is what? I don't get this.

"If P, then Q" means "If P is true, then Q is true."

Most logic texts list common interpretations of "if P, then Q." Compare these to the sentences in parts (a) through (f) to find those equivalent to "if P, then Q."

2. Using predicates S(x) for "x is smart" , M(x) for "x makes a lot of money" , formalize the following satements.

a) Some smart people make a lot of money.
b) Not all smart people make a lot of money.
c) People who are smart make a lot of money
d) Only smart people make a lot of money.
e) There are people who are not smart but make a lot of money.


I don't get this one either. How am I supposed to use S(x) and M(x) here?


Thank you!

You are expected to use quantifiers and the predicates S(x) and M(x) to form quantified statements.

For example "If no one were smart, then no one would make any money," is formalized as

$$\neg \exists x \;S(x) \Rightarrow \neg \exists x \;M(x)$$

--Elucidus

 

[XodoX]

Thanks. I got 1. now, but still not sure about 2.

Let's take a).

S(x) is smart people and M(x) is the money. Like the p and q in 1. M(x) never changes. it's always " make a lot of money".
Would a) be "S(x) V M(x)" ?

 

[lanedance]

Thanks. I got 1. now, but still not sure about 2.

Let's take a).

S(x) is smart people and M(x) is the money. Like the p and q in 1. M(x) never changes. it's always " make a lot of money".
Would a) be "S(x) V M(x)" ?

for a given x (a person)
if x is smart, then S(x) is true, otehrwise false
if x makes a lot of money, then M(x) is true, otherwise false

so for the c) which is probably easiest, you have
c) People who are smart make a lot of money
$$S(x) \rightarrow M(x)$$

by the way I'd be intersted to see what you got for 1)

 

[Elucidus]

Xodox:

Your interpretation for part (a) $S(x) \vee M(x)$ is not quantified yet. Also you are using a disjunction (or) which indicates either x is smart or makes a lot of money.

You need to use either $\forall x$ or $\exists x$ for each predicate form.

lanedance:

You also need quantifiers. But keep in mind there is a difference between

$\forall z P(z) \rightarrow \forall z Q(z) \text{ and } \forall z (P(z) \rightarrow Q(z))$.

--Elucidus

 

[lanedance]

Hi Ecludius

I'm not sure i understand?

I would take the shorthand
$$S(x) \rightarrow M(x)$$

to mean
$$\forall x \in X, S(x) \rightarrow M(x)$$

what do you mean by
$$\forall z P(z) \rightarrow \forall z Q(z)$$ ?

 

[Elucidus]

I was saying not to mistakingly interpret this as $\forall x, S(x) \rightarrow \forall x, M(x)$ which means something else. I've seen people have this confusion. I was using P and Q in the generic sense as an example.

--Elucidus

 

[evagelos]

1. Mark as true or false. The implication "If P, then Q " is equivalent to:

a) P is necessary for Q
b) Q is sufficient for P
c) P is sufficient for Q
d) Q is necessary for P
e) P only if Q
f) Q only if P

It seems like there's missing something..If P is what? I don't get this.



2. Using predicates S(x) for "x is smart" , M(x) for "x makes a lot of money" , formalize the following satements.

a) Some smart people make a lot of money.
b) Not all smart people make a lot of money.
c) People who are smart make a lot of money
d) Only smart people make a lot of money.
e) There are people who are not smart but make a lot of money.


I don't get this one either. How am I supposed to use S(x) and M(x) here?


Thank you!

For the 1st part we have:

a) false
b) false
c) true
d) true
e)false
f) true

For the 2nd part we have:

a) $$\exists x[ S(x)\wedge M(x)]$$

b)$$\neg\forall x[ S(x)\Longrightarrow M(x)]$$ which is equivalent to:

$$\exists x [ S(x)\wedge\neg M(x)]$$

c) $$\forall x[ S(x)\Longrightarrow M(x)]$$

d)$$\forall x[ S(x)\Longleftrightarrow M(x)$$

e)$$\exists x[ \neg S(x)\wedge M(x)]$$

 

[honestrosewater]

a) false
b) false
c) true
d) true
e)false
f) true

To me, "P only if Q" means that Q is necessary for P, i.e., if Q is not true, then P is not true, i.e., ~Q --> ~P. What statement is this equivalent to?

a) $$\exists x[ S(x)\wedge M(x)]$$

b)$$\neg\forall x[ S(x)\Longrightarrow M(x)]$$ which is equivalent to:

$$\exists x [ S(x)\wedge\neg M(x)]$$

c) $$\forall x[ S(x)\Longrightarrow M(x)]$$

d)$$\forall x[ S(x)\Longleftrightarrow M(x)$$

e)$$\exists x[ \neg S(x)\wedge M(x)]$$

I take "Only smart people make a lot of money" to mean that if you make a lot of money, then you are smart, but I don't think it implies that if you are smart, then you make a lot of money. I.e., it isn't saying that all smart people make money, just that all people that make money are smart.

I agree with you on the others. Are you the OP or someone else? We aren't supposed to simply give complete answers here.

 

[regor60]

"f" is definitely false, that's very straightforward

 

[honestrosewater]

(e) and (f) have the same logical form. They should be equally straightforward.