Let N be the set of positive integers and let X = {(m,n) in N x N: m < n}. For each statement below either?

Question

give a proof or state and give the negation:
(a) (2,4) in X
(b) (3,1) in X
(c) All m in N, Some n in N, (m,n) in X,
(d) Some n in N, All m in N (m,n) in X

Answer 1

(a) (2,4) in X
true, since 2<4.
(b) (3,1) in X
false, because 3>1
(c) All m in N, Some n in N, (m,n) in X,
true,
let m in N, then define n=m+1, then (m,n)=(m,m+1) is in X.
(d) Some n in N, All m in N (m,n) in X
false
assume that there is an n, such that
for every other m, (m,n) is in X,
but this means that there is an n, such that m

Answer 2

a) True. (2,4) is in X because 2<4.
b) False. (3,1) is *not* in X, because 3 is not less than 1.
c) True. For all m in N, let n = m+1. Then m d) False. There doesn't not exist an n such that for all m in N, (m,n) is in X. Proof: For any possible n, let m = n+1. Then (m,n) is not in X, because n+1>m.

Answer 3

good ?