Identifying the truth value of quantified statements

Question

I need help properly identifying the truth values of the following quantified statements:

i. Q(0, -3, 1), where Q(x,y,z) := "The numbers x-y, x+2y, and $z^2+1$ are even."

ii. P(Angela) and P(Anderson), where P(X) := "The name X contains at least two vowels."

I understand that the first statement is a universal truth statement and I would have to check that the truth value of $(x-y)\wedge (x+2y) \wedge (z^2 +1)$ is true. As for the second statement, I think that I must prove that that the set consisting the letters of the name "Angela Anderson" has at least one true value.

I think that I'm just having some trouble formatting my answer into the "proper" way of proving the truth values. Can anyone help? Thank you!

Answer 1

i. $$Q(0, -3, 1),\tag1$$ where Q(x,y,z) := "The numbers x-y, x+2y, and $z^2+1$ are even."

I understand that the first statement is a universal truth statement

This is a straightforward plug-and-check exercise; you should treat it as just another elementary mathematics question. $(1)$ is not a "universal" statement; it is easier to make plain sense of the object/sentence/statement $(1)$ when it is not kept at an distance by abstract adjectives. How old is the shepherd?

and I would have to check that the truth value of $(x-y)\wedge (x+2y) \wedge (z^2 +1)$ is true.

Simply: "....check that the truth value of (x−y)∧(x+2y)∧(z2+1) is true."

But, in fact, (x−y)∧(x+2y)∧(z2+1) is not a sentence, so how can it have a truth value? It does not make sense to say that x-y, which equals $3,$ is true, or that it is false. Just try the suggestion in my first sentence above.

ii. $$\text {P(Angela) and P(Anderson),}\tag2$$ where P(X) := "The name X contains at least two vowels."

As for the second statement, I think that I must prove that that the set consisting the letters of the name "Angela Anderson" has at least one true value.

No, this doesn't require considering text strings or sets of letters. Again, just confidently plug and check, noting that statement $(2)$ is true precisely when both conjuncts (its two parts that are left and right of "and") are true.

Answer 2

The first statement is not universal. Replace $x,y,z$ by $0,−3,1$ in the statement and see if the three (because of the "and" int the statement) integers you obtain are all even, or if one of them (or several) is odd.

As for the second statement, your thoughts are ununderstandable (it means nothing for a set of letters to have "at least one truth value"). What you must look at is whether each of the two (because of the "and" in the statement) words "Angela" and "Anderson" contains at least two vowels.