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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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Let A = R { 0 } , the set of all nonzero real numbers, and consider the following relations on A × A . Decide in each case whether R is an equivalence relation, and justify your answers.

( a , b ) R ( c , d ) if and only if a d = b c .

( a , b ) R ( c , d ) if and only if a b = c d .

( a , b ) R ( c , d ) if and only if a 2 + b 2 = c 2 + d 2 .

( a , b ) R ( c , d ) if and only if a b = c d .

(a)

To determine

Whether the relation R on set A×A defined by (a,b)R(c,d) if and only if ad=bc is an equivalence relation or not.

Explanation

Given information:

A={0}, the set of all non-zero real numbers. Relation R is defined on set A×A.

The relation R defined by (a,b)R(c,d) if and only if ad=bc.

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:

Let (a,b)A×A

As multiplication in set of real numbers is commutative implies that ab=ba

Thus (a,b)R(a,b).

Therefore the relation R is reflexive.

2. Symmetric Property:

Let (a,b),(c,d)A×A

If (a,b)R(c,d) then ad=bc.

bc=ad

As multiplication in set of real numbers is commutative implies that bc=cb and ad=da.

cb=da

(c,d)R(a,b)

Therefore the relation R is symmetric

(b)

To determine

Whether the relation R on set A×A defined by (a,b)R(c,d) if and only if ab=cd is an equivalence relation or not.

(c)

To determine

Whether the relation R on set A×A defined by (a,b)R(c,d) if and only if a2+b2=c2+d2 is an equivalence relation or not.

(d)

To determine

Whether the relation R on set A×A defined by (a,b)R(c,d) if and only if ab=cd is an equivalence relation or not.

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