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Name: Perm number:

Final exam – take-home part

Due date: Monday, June 8th

(1) (25 points) A partial order is a binary relation that is reflexive, antisymmetric, and transitive. Check if the following relations are partial orders:

• aRb ⇔ a|b defined in the set Z

• aRb ⇔ a ≤ b defined in the set N

• ARB ⇔ A ⊂ B defined in the set P (X) for some nonempty set X

• aRb ⇔ a ≤ b ≤ a + 1 defined in the set Z

• (a, b)R(c, d) ⇔ (a = c ∧ b < d) ∨ (a < c ∧ b = d) defined in the set N × N

(2) (25 points) For a set X with a relation of partial order R the element x ∈ X is called the least element of X if, for all y ∈ X, xRy. Prove that in a partially ordered set there exists at most one least element.

(3) (25 points) A linear order is a binary operation that is antisymmetric, transitive, and total. A well-order relation is a linear order with the property that every non-empty subset S has a least element. Check if the following sets are linearly or well- ordered:

• Z

• Q

• R

• {(1n, 1] : n ∈ N}

• P (X) for any set X

(4) (25 points) Suppose every nonempty subset of a partially ordered set has a least element. Does it follow that this set is well-ordered?

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