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45 Cards in this Set
- Front
- Back
- 3rd side (hint)
Inductive Reasoning
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The process reasoning that a rule or statement is true because specific cases are true.
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specific cases |
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Conjecture
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A statement you believe to be true based on inductive reasoning.
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statement you believe |
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Counterexample
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An example that proves the conjecture false.
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example proves wrong |
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Example of Conjecture
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The product of a number squared is always an even number.
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number squared always even |
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Example of Counterexample
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The product of a number squared is always an even number.
A counterexample would be 3. (3 x 3 = 9) |
3*3=9 |
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Conditional Statement
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A statement that can be written in the form "if p then q"
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if p then q |
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Hypothesis (geometry definition)
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The part "p" of a conditional statement, following the word "if"
p (hypothesis) -> q |
p follows the word if |
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Conclusion (geometry definition)
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The part "q" of a conditional statement, following the word "then"
p -> q (conclusion) |
q follows the word then |
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Truth Value
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Tells whether a conditional statement is true (T) or false (F). It is false only when the hypothesis is true and conclusion is false. The truth value is true if the conclusion is true.
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Negation
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The negation of statement "p" is written as "~p". The negation of a true statement is false and the negation of a false statement is true.
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Converse of Conditional Statement
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The statement formed by exchanging the hypothesis and conclusion. In other words, you flip them.
q -> p |
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Inverse of Conditional Statement
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The statement formed by negating the hypothesis and the conclusion. You do this by proving the hypothesis and conclusion wrong.
~p -> ~q |
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Contrapositive of Conditional Statement
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The statement formed by both exchanging and negating the hypothesis and conclusion.
~q -> ~p |
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Logically Equivalent Statements
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Related Conditional Statements that have the same truth value. A conditional and its contrapositive are equivalent, and so are the converse and inverse.
Ex: If the converse (q -> p) and the inverse (~p -> ~q) are both false, they are logically equivalent. |
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Example of Negation
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(hypothesis) If an animal has four paws, (conclusion) it is a dog.
A cat also has four paws, so this statement is false. |
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Deductive Reasoning
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The process of using logic to draw conclusions from given facts, defintions, and properties.
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Law of Detachment
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The Law of Detachment is one valid form of deductive reasoning.
If p -> q is a true statement, then p is true and q is true. |
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Law of Syllogism
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Another form of valid deductive reasoning. This method allows you to draw conclusions from two conditional statements when the conclusion of one is the hypothesis of the other.
If p -> q and q -> r are true statements, then p -> r is a true statement. |
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Example of using the Law of Detachment in Conditional Statements
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Given: If you read ten books, then you get a reward. Tom gets a reward.
Conjecture: Tom gets a reward. The conjecture matches the conclusion, but that does not mean the hypothesis is true. Tom could have gotten a reward for something else. |
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Biconditional Statement
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A statement that can be written in the form "p if and only if q." This means "if p, then q" and if "if q, then p."
p <--> q means p -> q and q -> p |
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Definition (geometry definition)
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A statement that describes a mathematical object and can be written as a true biconditional.
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Example of Biconditional
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(hypothesis [p]) A polygon is a square if and only (conclusion [q]) if it has four congruent sides.
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Definition
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A statement that describes a mathematical object and can be written as a true biconditional. A good definition can be used forwards and backwards. For example, if a figure is a quadrilateral, then it is a four sided polygon. If a figure is a four sided polygon, it is a quadrilateral.
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Proof (geometry definition)
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An arguement that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true.
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Addition Property of Equality
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If a = b, then a + c = b + c
EX: a = 5 b = 5 (a) 5 + 10 = (b) 5 + 10 c = 10 |
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Subtraction Property of Equality
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If a = b, then a - c = b - c
EX: a = 5 b = 5 (a) 5 - 2 = (b) 5 - 2 c = 2 |
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Multiplication Property of Equality
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If a = b, then ac = bc
EX: a = 5 b = 5 (a) 5 x 10 = (b) 5 x 10 c = 10 |
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Division Property of Equality
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If a = b, and c =/= 0, then a/c = b/c
EX: a = 20 b = 20 (a) 20/10 = (b) 20/10 c = 10 |
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Reflexive Property of Equality
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a = a
(Technically, it's the same point/angle) EX: 5 = 5 |
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Symmetric Property of Equality
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If a = b, then b = a
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Transitive Property of Equality
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If a = b and b = c, then a = c
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Substitution Property of Equality
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If a = b, then b can be substituted for a in any expression
EX: If b = 5x, and a = 2b <------- (b = 5x), then you can substitute "b" into "a". So "a" would equal 2(5x) or 10x. |
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Properties of Equality
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Algebraic properties that are used to solve algebraic proof.
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Symmetric Property of Congruence
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If figure A is congruent to figure B, then figure B is congruent to figure A.
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Reflexive Property of Congruence
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Figure A is congruent to figure A
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Transitive Property of Congruence
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If figure A is congruent to figure B and figure B is congruent to figure C, then figure A is congruent to figure C.
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Theorem
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Any statement that you can prove.
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Linear Pair Theorem
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If two angles form a linear pair, then they are supplementary.
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Congruent Supplements Theorem
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If two angles are supplementary to the same angle, then the two angles are congruent.
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Two-Column Proof
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A way of solving a proof in which you put the steps of the proof in the left column and the matching reason for each step in the right column.
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Right Angle Congruence Theorem
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All right angles are coungruent.
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Congruent Complements Theorem
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If two angles are complementary to the same angle, then the two angles are congruent.
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Flowchart Proof
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A way of solving a proof that uses boxes and arrows to show the structure of the proof.
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Vertical Angles Theorem
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Vertical angles are congruent.
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Theorem 2-7-3
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If two congruent angles are supplementary, then each angle is a right angle.
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