Understanding Logical Statements: P(x) and Q(x) in Z | Examples and Explanation

[Demonoid]

I'm a little stuck with these bad boys:

Let P(x) be the assertion “x is odd”, and let Q(x) be the assertion “x is twice an integer.” Determine whether the following statements are true:

1. (Vx ∈ Z)(P(x) ⇒ Q(x))
2. (Vx ∈ Z)(P(x)) ⇒ (Vx ∈ Z)(Q(x))

My attempt:

I don't get the statement at all; if x is odd then 2x ? maybe if x is odd then 2x is even ?
Is that what they're trying to say ?

and for me the quantifiers look the same.
(Vx ∈ Z) a-> b is same as (Vx ∈ Z) a -> (Vx ∈ Z)b

I don't understand the question at all.

 

[chiro]

I'm a little stuck with these bad boys:

Let P(x) be the assertion “x is odd”, and let Q(x) be the assertion “x is twice an integer.” Determine whether the following statements are true:

1. (Vx ∈ Z)(P(x) ⇒ Q(x))
2. (Vx ∈ Z)(P(x)) ⇒ (Vx ∈ Z)(Q(x))

My attempt:

I don't get the statement at all; if x is odd then 2x ? maybe if x is odd then 2x is even ?
Is that what they're trying to say ?

and for me the quantifiers look the same.
(Vx ∈ Z) a-> b is same as (Vx ∈ Z) a -> (Vx ∈ Z)b

I don't understand the question at all.

Intuitively they both look like they are false, but can you remind me what the condition is for implication in terms of union and intersection of sets?

 

[eabod]

I'm a little stuck with these bad boys:

Let P(x) be the assertion “x is odd”, and let Q(x) be the assertion “x is twice an integer.” Determine whether the following statements are true:

1. (Vx ∈ Z)(P(x) ⇒ Q(x))
2. (Vx ∈ Z)(P(x)) ⇒ (Vx ∈ Z)(Q(x))

My attempt:

I don't get the statement at all; if x is odd then 2x ? maybe if x is odd then 2x is even ?
Is that what they're trying to say ?

and for me the quantifiers look the same.
(Vx ∈ Z) a-> b is same as (Vx ∈ Z) a -> (Vx ∈ Z)b

I don't understand the question at all.

I doubt if my response below says anything that your textbook/professor hasn't said, so your misunderstanding seems likely to be a symptom of a lack of reading/listening (and thinking about what you've read/heard). Learning to read/listen to mathematics takes work, but is invaluable (even outside of mathematics). In my explanations below, I recommend that whenever I ask a question, you refrain from reading the next sentence until you've tried to answer the question on your own.

I assume Z represents the set of integers; i.e. Z = {...-2, -1, 0, 1, 2, ...}.

Statement 1. should be read "for all x in the integers, if x is odd, then x is twice an integer". Is that true? Intuitively, it should be clear that the given statement is asking whether every odd integer is twice another integer, so to check it's truth you need to either prove it or find a counterexample. You might want to start by checking a few odd integers to see if they're twice another integer.

Although statement 1 is easy to interpret, the important thing we should be able to do is parse the statement to check for validity. This will be especially important in statement 2.

Statement 2. should be read "If, for all x in the integers, x is odd; then for all x in the integers, x is twice an integer". When beginning to parse this statement, the first thing to notice is that the sub-statements (Vx ∈ Z)(P(x)) and (Vx ∈ Z)(Q(x)) are separated by an implication arrow, so we essentially have a A ⇒ B type statement. How do we check if such a statement holds? Well, a statement A ⇒ B is true in all cases except one: when A is true yet B is false. So we need to check the validity of A and B separately. If we happen to find that A is false, then we don't even have to worry about B because the single case where A ⇒ B can be false is impossible. Is (Vx ∈ Z)(P(x)) true? What does that statement mean? It means "for every x in the integers, x is odd". Is that true? If not, you're done. If so, proceed to checking what (Vx ∈ Z)(Q(x)) means and determining its truth value.