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Understanding Properties of Equality in Mathematics

Learn about reflexive, symmetric, and transitive properties of equality, essential for algebra and geometric proofs. Set theory, equivalence relations, and examples demonstrate the significance of these properties.

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Understanding Properties of Equality in Mathematics

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  1. This section we will review the properties ofequalitythat 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. …\GeoSec02_04.ppt

  2. 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. …\GeoSec02_04.ppt

  3. 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. …\GeoSec02_04.ppt

  4. This represents a relation. A a1 a3 a2 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. In the case to the left, a1= a1. For example; 5 = 5 or x = x …\GeoSec02_04.ppt

  5. A a c b 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 …\GeoSec02_04.ppt

  6. Relation of AtoC c2 a1 b2 c3 a3 c1 b3 a2 b1 The third property requires three sets A, B, and C, and the relationsAtoB, and BtoC. With these connections you can establish a new relation of AtoC. This is the transitive property. Think of transitivity as moving through sets. Relation of AtoB Relation of BtoC The relation from AtoB is different than the relation from BtoC as well as the relationAtoC. …\GeoSec02_04.ppt

  7. Reflexive Property For every number a, a=a. Symmetric Property For all numbers a and b, ifa=b,thenb=a. Transitive Property For all numbers a, b, and c, ifa=b and b=c, thena=c Addition and Subtraction Properties For all numbers a, b, and c, ifa=b, thena + c=b + c and a - c=b - c . Multiplication and Division Properties For all numbers a, b, and c, ifa=b, thena•c=b•c and ,c 0. a/c=b/c Substitution Property For all numbers a and b, ifa=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 …\GeoSec02_04.ppt

  8. Property Segments Angles Reflexive PQ=PQm 1 =m 1 Symmetric IfAB=CD, thenCD=AB.ifmA=mB, then mB=mA Transitive IfGH=JK, JK=LM, thenifm1 =m 2 and m 2 =m 3, GH = LM. then m1 =m 3. …\GeoSec02_04.ppt

  9. Summary 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. …\GeoSec02_04.ppt

  10. END OF LINE …\GeoSec02_04.ppt

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