   Chapter 1.7, Problem 19E

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# For each of the following relations R defined on the set A of all triangles in a plane, determine whether R is reflexive, symmetric, or transitive. Justify your answers.a. a R b if and only if a is similar to b .b. a R b if and only if a is congruent to b .

a)

To determine

Whether the given relation R on A is reflexive, symmetric or transitive.

Explanation

Given Information: A is the set of all triangles in a plane and a relation R on A is defined by aRb if and only if a is similar to b.

Formula Used:

(1) A relation R on a nonempty set A is reflexive if the following property is satisfied:

xRx for all xA.

(2) A relation R on a nonempty set A is symmetric if the following property is satisfied:

If xRy, then yRx.

(3) A relation R on a nonempty set A is transitive if the following property is satisfied:

If xRy and yRz, then xRz.

Explanation:

Let a,b,cA.

1. aRb,sinceeverytriangleissimilartoitself.

So, R is reflexive.

2. aRbaissimilartob.Measureofeveryangleofaissameasmeasureofeveryangleofb.bissimilartoa.bRa.

So, R is symmetric

b)

To determine

Whether the given relation R on A is reflexive, symmetric or transitive.

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