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1.1 Sets and SubsetsObjectives:Learn and use introductory set notation to express sets, elements, and subsets.Use a Venn Diagram to visualize a setVocabularySymbols/ExamplesSET set-builder notation:ELEMENTEQUALEQUIVALENTSUBSETEMPTY SET (NULL SET)UNIVERSAL SETGiven: A=1,3,5, B=2,4,6, C=1,2,3,4,5, D={2,4,6}Give a yes or no answer:Is 2∈A?Is B?C?Is A?C?Is A?C?Does D=B?Is A equivalent to B? Is A equal to B?Is 3∈C?Venn Diagrams: They represent sets visually using overlapping circles.923925-1270Example: A=B=U(**universal set)=10382257620Example: A=B=C=U=103822590805Example:A=B=C=U=1.2: Set OperationsObjective:To identify and perform the basic operations on sets.To use Venn diagrams to illustrate the basic operations on sets.Union: The union of two sets is the set combining all the elements of the given sets. The union of set A and B is all the elements in either set A OR set B. “AUB” *binary operationExamples: If T={1,2,3} and P={4,5,6},If Y={1,3,5} and Z={7,9,10},If A={1,2} and B={2,3}***numbers don’t repeatIf C={1,2,3} and D={1,2,3}2857524765Intersection: The intersection of two sets is the set containing the elements that belong to both sets. The intersection of set A and B is all the elements in both set A AND B. “A?B” *binary operationExamples:If T={1,2,3} and P={4,5,6},If Y={1,3,5} and Z={7,9,10},If A={1,2} and B={2,3}If C={1,2,3} and D={1,2,3}Complement: The complement of a set is the set of all elements in the universal set that are not in the original set. The complement of M is M’. U-M=M’Examples:If U={1,2,3,4,5,6,7,8,9}and C={1,2,3}If U={0,1,2,3,4,…}and C={1,3,5,7,...}A’ is shaded.Order of Operations for Sets:Parentheses ComplementsUnionsintersectionsPractice:Let U={1,2,3,4,5,6,7}. A={1,2,3,4} and B={1,3,5,7}.Quick-reference:7620043180Find and diagram AUB.Find and diagram A?B.Find and diagram A’.Find and diagram A’?B.Find and diagram (A?B)’.Find and diagram A-B.Find and diagram B-A.1.3: Undefined Terms and DefinitionsGood or Bad? You be the judge:Space is the set of all points.A square is a regular polygon with four sides.A baboon is a monkey.Snapchat is a bunch of picturesA fingernail is a nail on a finger.Wood is the material trees are made out of.A schism divides people.Characteristics of a good definition:1. Clear2. Useful3. Precise4. Concise5. ObjectiveTry to define the following words:BagSackBibleGod’s WordWindAir ***SOME IDEAS ARE UNDEFINED! WE MUST ACCEPT THEM ON FAITH!Geometry’s undefined terms:3810045720This is the crazy thing: we cannot define these terms! We just must believe in the world of geometry that these things exist. Yes, that can be annoying, but many things behave this way.1.4: A Framework with DefinitionsObjective:Define keystone geometric concepts for the year.Undefined Terms DefinitionsPostulates (assumed relationship)Theorems (proven relationship)Vocabulary WordDefinition/ExampleCollinear pointsDef: Points P, Q, and R are collinear.2693670-538480Noncollinear pointsDef:Points X, Y, and Z are noncollinear.2960370-581025Concurrent linesDef:Lines l, n, and m are concurrent. They intersect at point p.3084195-590550Coplanar pointsDef:3084195-392430Points A, B, and C are coplanar. They’re all in plane N.Coplanar lines274002521590Def:Lines n and m are coplanar.Parallel lines3007995132080Def:Lines AB and CD are parallel.Perpendicular lines269367060325Def:Intersecting linesDef: 2626995-281305Skew linesDef:2093595-946150Parallel planes2284095146050Def:Perpendicular planesDef:2893695-1283335Challenge: Are there such things as skew planes?Example: Use the diagram to the right to answer the following:Name the lines that contain point G.4076700-1270Name the lines that are concurrent at point B.Name all lines shown.Name two planes.Name a pair of skew lines.Name a pair of parallel lines.Name 3 noncollinear points.Name 2 coplanar lines.Name the intersection of plane K and plane L.1.5-1.6 Euclid“There is no royal road to geometry…” Important points about Euclid: Qualities of an ideal postulate system:1. Consistent2. Independent3. CompleteNo ideal system can be constructed. Because man would tend to boast of a perfect system, God has limited man’s logic and reason. In 1931, the modern mathematician Kurt Godel proved mathematically that ideal postulate systems are impossible. So then does geometry crumble? No. God is the foundation for geometry. God created geometry and He is consistent, independent, and complete. The holdup is that no system is ever complete. We can never possibly know everything there is to know about a system of geometry. In itself, though, that makes life beautiful. Euclid’s Five Postulates (Axioms) Stated in “The Elements”1. Any two points determine a unique line.2. Any straight line segment can be extended indefinitely in a straight line.3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.4. All right angles are congruent.5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.Proofs: Series of logical deductions starting with known facts and ending with a new statement. Think of yourself as a geometric lawyer. You must provide evidence that the theorem stated belongs to our structure of geometry –the 5 Euclidean axioms. Proposition 1.1: Euclid’s first proof!1.7 Sketches and ConstructionSketch: making a neat freehand pictureDraw: using any tools you wish, including rulers and protractors, but freehand is forbiddenConstruct: requires drawing using only a straightedge and compassSketch a lineDraw a lineConstruct a lineSketch a circleDraw a circleConstruct a circleSketch a 2 inch segmentDraw a 2 inch segmentConstruct a 2 inch segmentDaffodil construction! (instructions on page 35 of book) ................
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