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Download Lesson 86: Greater Than, Trichotomy and Transitive Axioms
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Transcript
Lesson 86: Greater Than, Trichotomy and Transitive Axioms, Irrational Roots One number is said to be greater than another number if its graph on the number line lies to the right of the graph of the other number. Thus, we see that -1 is greater than -4, because the graph of -1 lies to the right of the graph of -4. Graphing inequalities on a number line afford practice with the concept of greater than and also allows us to remember the definitions of the domain and of the subsets of the set of real numbers. Recall that the domain of an equation or inequality is the set of permissible replacement values of the variable. Example: Graph: x < 3; D = {Positive Integers} Answer: See Example Example: Graph: x > -3; D = {Reals} Answer: See Example Some fundamental concepts of mathematics are difficult to remember because they are so self-evident. The trichotomy axiom and the transitive axiom are good examples. The trichotomy axiom can be demonstrated by having someone write a number on a piece of paper. Then let that person turn the paper over and write a number on the other side. There are then exactly three possibilities. 1. The second number is the same number as the first number. 2. The second number is greater than the first number. 3. The second number is less than the first number. These statements are self-evident but are not trivial. They tell us that the real numbers are arranged in order, and thus we can say that the real numbers constitute an ordered set. l we say that these three statements form the trichotomy axiom. Trichotomy comes from the Greek work trikha, which means “in three parts”. The transitive axiom also has three parts. It is also self-evident but not trivial. If Arthur is larger than Billy and Billy is larger than Susan, then Arthur is larger than Susan. The same statement can be made using smaller than or the same size as instead of larger than. Real numbers are just like people in this respect, because this thought also applies to real numbers. We state both of these axioms formally in the following box. Axioms For any real numbers, a, b, and c: Trichotomy Axiom Exactly one of the following is true: a<b, a = b, or a>b Transitive Axiom If a>b and b>c, then a>c If a<b and b<c, then a<c If a=b and b=c, then a=c Example: Graph: x ≤⁄ -4; D = {Negative Integers} Answer: See Example Example: Graph: -x + 4 > 2; D = {Reals} ⁄ Answer: See Example Example: Graph -x – 4 ≥⁄ -2; D = {Negative Integers} Answer: See Example Some equations that describe the points of intersection of a line and a circle cannot be factored, because the solutions to these equations are irrational numbers. These solutions can be found quickly and easily by using the quadratic formula. Many people always use this formula when solving real-life quadratic equations, because so many of these equations cannot be solved by factoring. Example: Solve 2 2 x+y=9 y–x=1 Answer: (-1/2 + √17/2, ½ + √17/2) (-1/2 − √17/2, ½ − √17/2) HW: Lesson 86 #1-30