**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 *R*^{2} = {(*a*, *a*), (*b*

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