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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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Let l 1 and l 2 be lines in a plane. Decide in each case whether or not R is an equivalence relation, and justify your decisions.

l 1 R l 2 if and only if l 1 and l 2 are parallel.

l 1 R l 2 if and only if l 1 and l 2 are perpendicular.

(a)

To determine

Whether R is an equivalence relation or not.

Explanation

Given information:

l1 and l2 be two lines in a plane. R be a relation defined by l1Rl2 if and only if l1 is parallel to l2.

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:

Let l1 and l2 be two lines in a plane.

Let R be a relation defined by l1Rl2 if and only if l1 is parallel to l2.

1. Reflexive property:

Let l1 be line in a plane.

By using, every line is parallel to itself.

Thus l1 is parallel to l1 itself.

Hence l1Rl1

Therefore the relation R is reflexive

(b)

To determine

Whether R is an equivalence relation or not.

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