   Chapter 1.7, Problem 13E

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# Consider the set ℘ ( A ) − { ϕ } of all nonempty subsets of A = { 1 , 2 , 3 , 4 , 5 } . Determine whether the given relation R on ℘ ( A ) − { ϕ } is reflexive, symmetric or transitive. Justify your answers.a. x R y if and only if x is subset of y .b. x R y if and only if x is a proper subset of y .c. x R y if and only if x and y have the same number of elements.

a)

To determine

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

Explanation

Given Information:

(A){ϕ} is the set of all nonempty subsets of A={1,2,3,4,5} and a relation R on a nonempty set (A){ϕ} defined by xRy if and only if x is a subset of y.

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){ϕ}

b)

To determine

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

c)

To determine

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

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