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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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In each of the following parts, a relation R is defined on the power set ( A ) of the nonempty set A . Determine in each case whether R is reflexive, symmetric, or transitive. Justify your answers.

a. x R y if and only if x y ϕ .

b. x R y if and only if x y .

a)

To determine

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

Explanation

Given Information:

(A) is the power set of the nonempty set A and a relation R on (A) is defined by xRy if and only if xyϕ.

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 x,y,z(A).

1. Let x=ϕ.

xx=ϕxRx

So, R is not reflexive

b)

To determine

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

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