* Inductive Reasoning –
* general conclusion based on a limited collection of specific observations * educated guesses
* Primary flaw – we cannot be sure the conclusion is always correct * Counterexamples -- show a conclusion reached through inductive reasoning is not true
* Deductive Reasoning –
* making a specific conclusion based on a collection of generally accepted assumptions. * There are no counterexamples
* Premises – undefined terms, definitions and postulates (or previously proven theorems) * Fallacy: A conclusion that does not necessarily follow from the premises.
* Proof by Negation – Indirect Proofs –
* Start with the Givens
* Assume the negation of the Conclusion/Proof
* The conclusion of the proof will be something that is wrong – contradicts another given. * So, what you want to prove must be true – the opposite of where you started. * Only works when what you want to prove has only two options – it either is or it isn’t. * Good when you are trying to prove that something is not true. Assume it is true and then prove that this makes no sense given what we are given.
2. Axiomatic System
* 4 Parts:
* Undefined Terms: Starting point of a system. Have intuitive meaning. * Definitions: Statements that give meaning to new terms that will be used in a system. * Axioms or Postulates: Statements about undefined terms and definitions that are accepted as true without verification or proof. * Theorems: Statements that can be proven by using definitions, postulates and the rules of deduction and logic. * Corollary: a theorem that is easy to prove as a direct result of a previously proved theorem. * Conditional Statement: a statement that implies a cause and effect (if p, then q) * Hypothesis: The “If” Clause
* Conclusion: the “Then” clause
* Many theorems are expressed as conditional statements * Converse – If q, then p. Not always a true statement * Inverse – If not p, then not q. Not always a true statement * Contrapositive – If not q, then not p. Always the same as the original statement * Biconditional statement – when a statement and its converse are always true 3. Points, Lines, and Planes –
* Points: represented by dots. Infinitely small. Determines position but has no dimension.
* Line – infinitely long straight string of points that extends forever in both directions. * Two lines are perpendicular if they intersect and form equal adjacent angles (which will be right angles). * There is exactly one line perpendicular to a given line passing through a given point on the line (P1.17) * There is exactly one line perpendicular to a given line passing through a given point not on that line (P1.18). * Parallel – 2 lines that are coplanar and that never intersect * If a line (or segment) is a perpendicular bisector, then it is perpendicular to the segment and passes through the segment’s midpoint. * Divides the segment into two congruent segments.
* Every point on the line is equidistant to the segment’s endpoints. (T2.3) * If 2 points are each equidistant to the endpoints of a segment, then those two points determine the segment’s perpendicular bisector. * Concurrent: Lines that intersect in one and only one point.
* Plane: A set of points on a flat surface having 2 dimensions and extending without boundary. No plane contains all the points in space. * Space: the set of all points. Any set of points, lines, or planes in space is called a geometric figure. * Coplanar – 2 lines that lie on the same plane. The line determined by 2 points on a plane is also on the plane. * If any 2 points are on a plane and a line is drawn with them, then all points on that line are on the plane.
4. Basic Algebra...