Integration Algebra: Using Proof in Algebra

Section 2.4

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This section we will review the properties of equality that are useful for algebra in general and for

geometric proofs specifically.

We will specifically talk about the reflexive, symmetric, and transitive property of equality.

In this section we formalize their definition and use.

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Set theory deals with things (called elements or members) that we put in the same container.

Thus, the set of ALL cats does not have any dogs in that set.

The set of ALL house pets would include both cats and dogs, but not ALL cats and dogs are house pets.

It is important to understand what is and is not in a set.

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Mathematicians are always looking for new number systems that have specific relation properties.

One of the most desirable relation is called an equivalence relation.

An equivalence relation is defined are a relation that is reflexive, symmetric, and transitive.

The concept of the equality relation (i.e., = symbol) is an equivalence relation and as we continue in this course we will

find that congruence, , is an equivalence relation.

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An element in set A has an equality relation with itself. Specifically, it is equal to itself.

This property is called reflexive since it is a reflection of itself.

a2

a1a3

A In the case to the left, a1 = a1.

For example; 5 = 5 or

x = x

This represents a relation.

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If the order of the relation (in this case equality) does not matter, then the relation is called symmetric.

You can think of this as looking the same on both sides of the equality symbol.

In the case to the left, if a = b, then b = a

For example; if 5 = 2 + 3, then 2 + 3 = 5

b

ac

A

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The third property requires three sets A, B, and C, and the relations A to B, and B to C. With these connections you can

establish a new relation of A to C. This is the transitive property. Think of transitivity as moving through sets.

Relation of B to C

a2

a1a3

b2

b1

b3

Relation of A to B

c2

c1

c3

Relation of A to C

The relation from A to B is different than the relation from B to C as well as the relation A to C.

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Reflexive Property For every number a, a = a.

Symmetric Property For all numbers a and b, if a = b, then b = a.

Transitive Property For all numbers a, b, and c, if a = b and b = c, then a = c

Addition and Subtraction Properties

For all numbers a, b, and c, if a = b, then a + c = b + c and a - c = b - c .

Multiplication and Division Properties

For all numbers a, b, and c, if a = b, then a • c = b • c and ,c 0.

a/c = b/c

Substitution Property For all numbers a and b, if a = b, then a may be replaced

by b in any equation or expression.

Distributive Property For all numbers a, b, and c, a(b + c) = ab + ac

Properties of Equality for Real Numbers

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Reflexive PQ = PQ m 1 = m 1

Symmetric If AB = CD, then CD = AB. if m A = m B, then m B = m A

Transitive If GH = JK, JK = LM, then if m 1 = m 2 and m 2 = m 3,

GH = LM. then m 1 = m 3.

Property Segments Angles

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In this section we established the reflexive, symmetric, and transitive properties of equality that you will use throughout

the remainder of this course.

We briefly established that segments and angles, from an equality perspective, are also reflexive, symmetric, and

transitive.

We will use these properties in the next section to show that congruence is reflexive, symmetric, and transitive.

Summary

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