Let S ⊆ N, and for any c, d ∈ N, consider a relation R such that cRd if and only if there exists a ∈ S such that c + a = d Show that if R is a partial order, then (i) 0 ∈ S (ii) for any c, d ∈ S, c + d ∈ S.

Theorem: Let R ⊆ A × A be a relation. Then R is transitive if and only if R ◦ R ⊆ R. Prove Theorem: show that R is transitive if and only if R ◦ R ⊆ R. No handwritten please

Let T be the set {w = {0, 1}* ||w| ≤ 4}. Let R be the equivalence relation defined on T as follows: R = {(x, y) | x ≤T, yɛT, no(x) = = no(y)}, where no(r) represents the number of zeroes in the string x, and no(y) represents the number of zeroes in the string y. For example, (1011, 01) is a pair in R because the two strings 1011 and 01 have the same number of zeroes as each other. Every element in the set will appear in exactly one equivalence class and will be related to all elements in its class and not related to any elements outside of its class. What are the equivalence classes of T created by the relation R?

Let R be a relation and S its reverse. Show that R is injective if and only if S is well defined, and that R is surjective if and only if S is everywhere defined.