(a) Let ≺ be a strict partial order on the set A. Define a relation on A by letting a ≼ b if either a ≺ b or a = b. Show that this relation has the following properties, which are called the partial order axioms:
(i) a ≼ a for all a ∈ A.
(ii) a ≼ b and b ≼ a ⇒ a = b.
(iii) a ≼ b and b ≼ c ⇒ a ≼ c.
(b) Let P be a relation on A that satisfies properties (i)–(iii). Define a relation S on A by letting aSb if aPb and a ≠ b. Show that S is a strict partial order on A.
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