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#### Chapter 4

##### Section 4.1

1. a. Reflexive, symmetric, transitive. c. Reflexive, symmetric, transitive. e. Reflexive, symmetric, transitive. g. Irreflexive, transitive. i. Reflexive.

2. a. Symmetric. c. Reflexive, antisymmetric, and transitive. e. Symmetric.

3. a. The irreflexive property follows because (x, x) ∈ ∅ for any x. The symmetric, antisymmetric, and transitive properties are conditional statements that are always true because their hypotheses are false.

4. a. {(a, a), (b, b), (c, c), (a, b), (b, c)}.

c. {a, b)}.

e. {(a, a), (b, b), (c, c), (a, b)}.

g. {(a, a), (b, b), (c, c)}.

5. a. isGrandchildOf. c. isNephewOf. e. isMotherInLawOf.

6. isFatherOf ∘ isBrotherOf.

7. a. Let R = {(a, b), (b, a)}. Then R is irreflexive, and R2 = {(a, a), (b

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