FinalStudyingGeometry - Final Studying Geometry Logic...

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Final Studying: GeometryLogic Statements:Statement:Something that is either true or false, but not both CompoundStatement:When two or more statements, represented by the terms p and q, are put in asentence and are separated by logical connectors (“and” or “or”.) There are 2 types of compoundstatements, conjunctions and disjunctions. 1. Conjunction: Denoted as pΛq, which means p and q. (It is rainy and it is windy.) 2. Disjunction: Denoted by pVq, which means p or q. (It is rainy or it is windy.) Negation:Turns any statement into its opposite (true becomes false, false becomes true.) If p were astatement, the negation of p would be “not p.” Denoted as ~p. ConditionalStatement:An “If…, then…” statement using the terms p and q, symbolized by *p→ q*. P iscalled the premise, and q is called the conclusion. (If it is windy, then it is raining) Converse Statement:If q, then p. *q→ p* Inverse Statement:If ~p, then ~q . *~p→ ~q* Contrapositive Statement:If ~q, then ~p. *~q→ ~p* Contrapositive Statement is logically equivalent to Conditional Statement. ~q→ ~pis logically equivalent to p→ q DeMorgan’s Laws:~(pΛq) ~pV~qAND ~(pVq) ~pΛ~q(means logically equivalent!) We “distribute” the negation, turning the p to ~p, q into ~q, and Λ into V, or V into Λ. (Don’t forget toflip sign!) BiconditionalStatement:A compound statement formed by the conjunction of a conditional statementand its converse. If p, then q, and if q, then p. (p→ q)Λ(q→ p): Can be written 3 ways: 1. pq 2. p if and only if q 3. p iff q Note: A biconditional statement is also true when p and q have the same truth value (either TT or FF) Tautology:A statement that is already true, no matter whether p and q are not. Ex: (p→ q)V(q→ p). Contradiction:A statement that is always false, no matter whether p and q are not. Ex: (pVq)Λ(~pΛ~q)
p q pΛq pVq ~p ~q p→ q q→ p ~p→ ~q T T T T F F T T T T F F T F T F T T F T F T T F T F F F F F F T T T T T p q ~q→ ~p ~(pΛq) ~(pVq) pq (p→ q)V(q→ p) qΛ~q T T T F F T T F T F F T F F T F F T T T F F T F F F T T T T T (tautology!) F (contradiction!) Basic Terms:Collinear pointsare points that lie on the same line. • A is betweenB and C if A, B, and C are collinear and BA+AC=BC. • Aline segmentorsegmentis a set of two points called endpoints and all the points between them.(Symbol:)AB• The length of is represented by AB.AB• Arayis part of a line that strates at one point (called the endpoint) and extends endlessly in onedirection. (Symbol:) Opposite raysare two rays on the same line with a common endpoint but no other points in common. Congruentmeans equal in measure. (Symbol: ) • Themidpointof a line segment is a point on the segment that divides the segment into 2 congruentparts. • A segment bisectoris a line, ray, or segment that intersects the segment at its midpoint. • Anangleis a set of points consisting of 2 rays with a common endpoint (called the vertex) and no otherpoints in common.

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