Proving various relations are partial orders I am given these relations, in which I have to prove or disprove each and every one.
a. The relation $\trianglelefteq$ deﬁned on ℕ by a $\trianglelefteq$ b if a ≤ b²
b. The relation $\preceq$ deﬁned on ℤ by m $\preceq$ n if m ≤ n + 5.
c. The relation $\ll$ on the set of continuous functions deﬁned by f(x) $\ll$ g(x) if $\int^1_0f(x)dx \leqslant \int^1_0g(x)dx$
So, for a, I have the following: To prove reflexivity, it follows that a $\leqslant$ a², and therefore, a $\trianglelefteq$ a, so $\trianglelefteq$ is reflexive. For transitivity, for all a, b, and c in ℕ, if a ≤ b², and b ≤ c², then a ≤ c², and therefore the relation is transitive.
For b, I have that for reflexivity, m ≤ m+5, so the relation is reflexive. For transitivity,for all m,n, and p in ℤ, if m ≤ n + 5, and n ≤ p + 5, then m ≤ p + 5, so the relation is transitive.
For c, I wrote for reflexivity that since $\int^1_0f(x)dx \leqslant \int^1_0f(x)dx$, then it the relation is reflexive. For transitivity, if $\int^1_0f(x)dx \leqslant \int^1_0g(x)dx$, and $\int^1_0g(x)dx \leqslant \int^1_0h(x)dx$, then $\int^1_0f(x)dx \leqslant \int^1_0h(x)dx$, therefore, the relation is transitive.
As you can see, I left out antisymmetry, because I am not sure how to prove that. I think that my reflexive and transitive proofs are correct, but if anyone could help me out with these problems by helping me with the antisymmetry proofs and verifying/correcting my reflexive and transitive proofs, that would be great.

Check out the transitivity parts of your proof.
E.g. for a., we have $5\trianglelefteq 3$ and $3\trianglelefteq 2$ but not $5\trianglelefteq 2$.
If you already find that the given relation is not reflexive (or not transitive), then you're done as it cannot be partial order.
Finally, c. is indeed transitive, but we can have two different continuous functions $f,g$ on $[0,1]$ such that $f\ll g\ll f$. (You can choose one of them to be constant $0$.)
