Predicate Logic and Sets (Review Questions) I have a couple questions I have tried. These are simple homework questions I am doing for review for my 3rd year course in university. These are not for marks.

Predicate Question

1)

Let the predicate E(x,y) represent "Person x eats food y"

Let the predicate M(y) represent (Food y is a meat product"

a) Express the statement in predicate logic: "Someone is a vegetarian."

My Answer: Ǝx∀y E(x,y) ¬M(y)

New Answer: Ǝx∀y (E(x,y) => ¬M(y))

b) Express the statement in predicate logic:"Nobody (except maybe John) eats lasagna."

My Answer: ¬∀xƎy E(John,Lasagna)

New Answer: ∀x(E(x,lasagna) => (x = John))

Sets Question

2) Determine which of the following sets are finite or infinite. If set is finite show all elements in the set. If infinite show 5 elements in the set.

N represents Natural Numbers

a) A = {n ∈ N: ∀m ≤ n, n ≠ m^2}

My Answer: Infinite Set, A = {1, 4, 9, 16, 25} (not sure about this one)

New Answer: Infinite Set, A = {3, 5, 7, 9 , 11}

b) B = {n ∈ N : ∀m ≤ n, n = m^2}

My Answer: Infinite Set, B = {1, 4, 9, 16, 25}

New Answer: Infinite Set, B = {2, 4, 6, 8, 10}

c) C = {n^2 : n ∈ N and 2 &lt; n &lt; 6}

My Answer: Finite Set, C = { 9, 16, 25 }

d) D = {(n,m)∈ N x N: 1 &lt; n &lt; m &lt; 6}

My Answer: Finite Set, D = { (2,3), (3,4) , (4,5) }

New Answer: Finite Set, D = {(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)}

I appreciate any sort of help, if the answer is correct just reply about the ones that are not correct, thanks in advance.
I find it helps to have an intermediate step that says the same thing as the predicate, but in words rather than symbols. So for your first one ("Someone is a vegetarian"), after some massaging, I'd work with the words to make it "There exists a person, such that for all food, if that person eats it then it's not meat". Which would then be written as $\exists{x}\forall{y}\left(E(x,y)\implies\neg M(y)\right)$. Similarly for your second one, I'd go from "Nobody (except maybe John) eats lasagne" to "If someone eats lasagne, that person must be John" to "For all people, if that person eats lasagne, then that person is John" which gives $\forall{x} \left(E(x, lasagne) \implies (x=John)\right)$. Although there are a few equivalent versions, such as $\forall{x} \left(\neg E(x,lasagne) \lor (x = John) \right)$, which roughly translates to "For all people, it's either true that that person doesn't eat lasagne, or that person is John".

Similarly, for the second set of questions, try to describe the predicate.

$A = \left\{n \in \mathbb{N}: \forall {m \leq n}, n \neq m^2\right\}$ says that "A is the set of all natural numbers n, such that for all m less than or equal to n, n is not equal to m squared". Translate it into marginally more natural English and you get "A is the set of natural numbers that are not equal to the square of another number", which then gets to "A is the set of non-square numbers". Which must mean that $A = \left\{2, 3, 5, 6, 7, 8, 10, \dots \right\}$ which is an infinite set. How would that work for the rest of the questions?
Question 1  

I have a couple questions I have tried. These are simple homework questions I am doing for review for my 3rd year course in university. These are not for marks.

Predicate Question

1)

Let the predicate E(x,y) represent "Person x eats food y"

Let the predicate M(y) represent (Food y is a meat product"

a) Express the statement in predicate logic: "Someone is a vegetarian."

My Answer: Ǝx∀y E(x,y) ¬M(y)

b) Express the statement in predicate logic:"Nobody (except maybe John) eats lasagna."

My Answer: ¬∀xƎy E(John,Lasagna)

CHECKS:

1a  wrong (try if y is meaty then there is somebodty who won't eat it) 

1b wrong again  (try if somebody is eating lasanga then ithat person is john)

btw the answers you gave also don't look welformed.

and in principle giving an answer that is not wellformed is even worse than giving a wrong answer.

Sets Question

2) Determine which of the following sets are finite or infinite. If set is finite show all elements in the set. If infinite show 5 elements in the set.

N represents Natural Numbers

a) A = {n ∈ N: ∀m ≤ n, n ≠ m^2}

My Answer: Infinite Set, A = {1, 4, 9, 16, 25} (not sure about this one)

b) B = {n ∈ N : ∀m ≤ n, n = m^2}

My Answer: Infinite Set, B = {1, 4, 9, 16, 25}

c) C = {n^2 : n ∈ N and 2 &lt; n &lt; 6}

My Answer: Finite Set, C = { 9, 16, 25 }

d) D = {(n,m)∈ N x N: 1 &lt; n &lt; m &lt; 6}

My Answer: Finite Set, D = { (2,3), (3,4) , (4,5) }

CHECKS

2a what do you mean (try to describe it better, do you meean the infinite set of all quadrates , or the finite set containing  1, 4, 9, 16, and  25? , btw both answers are wrong)

2b again what do you mean (try to describe it better)

2c CORRECT  one point

2d you are missing some elements, hint (2,5) is one of them.

conclusion:

1 out of 6, you do need some more practice, Good luck

ps if you want to test new answers 

mark in your question

Second round

and give new answers

Good luck

2nd Marking

1a 2nd answer   Correct

1b 2nd answer   Correct

2a  there is a bigger problem here you still say the set is infinite and then say it has a limited number of elements. 

try to say what the elements of this set are in english, only after that translate ithat answer into a formula. DON't enumerate the set (do' t do something of the form: Infinite Set, A = {3, 5, 7, 9 , 11}

2b same problem, A = {a, b, c, d, e} is not an infinite set so where are you talking about? 

Conclusion
Getting there 

PS 1

Mauro's comment " We have to specify if $ 0 \in N$ or not. check your book different books have different conventions. so what is the convention in your class?