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Integration Algebra: Using Proof in Algebra Section 2.4

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Integration Algebra: Using Proof in Algebra 5/9/2015…\GeoSec02_04.ppt2 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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Integration Algebra: Using Proof in Algebra 5/9/2015…\GeoSec02_04.ppt3 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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Integration Algebra: Using Proof in Algebra 5/9/2015…\GeoSec02_04.ppt4 Visualize the different sets as being separated, and that the relation between the sets (the operation that ties one element in A to an element in B) is represented by an arrow from set A to set B.B. The arrow is directional, so just because you can go from set A to set B, does not mean you can go from set B to set A.A. Once you have established sets, the next step is to build relations between any given two sets. These relations are called binary relations (meaning two) since we are dealing with two sets at any given time. a2a2 a1a1 a3a3 b2b2 b1b1 b3b3 Relation of A to B

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Integration Algebra: Using Proof in Algebra 5/9/2015…\GeoSec02_04.ppt5 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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Integration Algebra: Using Proof in Algebra 5/9/2015…\GeoSec02_04.ppt6 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. a2a2 a1a1 a3a3 A In the case to the left, a 1 = a 1. For example; 5 = 5 or x = x This represents a relation.

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Integration Algebra: Using Proof in Algebra 5/9/2015…\GeoSec02_04.ppt7 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 = 5 b a c A

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Integration Algebra: Using Proof in Algebra 5/9/2015…\GeoSec02_04.ppt8 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 a2a2 a1a1 a3a3 b2b2 b1b1 b3b3 Relation of A to B c2c2 c1c1 c3c3 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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Integration Algebra: Using Proof in Algebra 5/9/2015…\GeoSec02_04.ppt9 Reflexive PropertyFor every number a, a = a.Symmetric PropertyFor all numbers a and b, if a = b, then b = a.Transitive PropertyFor all numbers a, b, and c, if a = b and b = c, then a = cAddition 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/ca/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 PropertyFor all numbers a, b, and c, a(b + c) = ab + ac Properties of Equality for Real Numbers

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Integration Algebra: Using Proof in Algebra 5/9/2015…\GeoSec02_04.ppt10 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, thenif m 1 = m 2 and m 2 = m 3, GH = LM.then m 1 = m 3. Property Segments Angles

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Integration Algebra: Using Proof in Algebra 5/9/2015…\GeoSec02_04.ppt11 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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Integration Algebra: Using Proof in Algebra 5/9/2015…\GeoSec02_04.ppt12 END OF LINE

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