   Chapter 1.7, Problem 14E

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# In each of the following parts, a relation is defined on the set of all human beings. Determine whether the relation is reflective, symmetric, or transitive. Justify your answers. x R y if and only if x lives within 400 miles of y . x R y if and only if x is the father of y . x R y if and only if x is a first cousin of y . x R y if and only if x and y were born in the same year. x R y if and only if x and y have the same mother. x R y if and only if x and y have the same hair colour.

(a)

To determine

Whether the relation R defined by xRy if and only if x lives within 400 miles of y is an equivalence relation or not.

Explanation

Given information:

A relation is defined on the set of all human beings.

The relation R defined by xRy if and only if x lives within 400 miles of y.

Formula used:

Definition of equivalence relation:

A relation R on a non-empty set A is an equivalence relation if the following conditions are satisfied for arbitrary x,y,z in A:

1. Reflexive property: xRx for all xA.

2. Symmetric Property: If xRy, then yRx.

3. Transitive Property: If xRy and yRz, then xRz.

Explanation:

1. Reflexive property:

Since x lives 0 miles of x itself.

Thus x lives within 400 miles of x itself.

Hence xRx.

Therefore the relation R is reflexive

(b)

To determine

Whether the relation R defined by xRy if and only if x is father of y is an equivalence relation or not.

(c)

To determine

Whether the relation R defined by xRy if and only if x is first cousin of y is an equivalence relation or not.

(d)

To determine

Whether the relation R defined by xRy if and only if x and y were born in the same year is an equivalence relation or not.

(e)

To determine

Whether the relation R defined by xRy if and only if x and y have same

mother is an equivalence relation or not.

(f)

To determine

Whether the relation R defined by xRy if and only if x and y have same hair color is an equivalence relation or not.

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