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54 Cards in this Set

  • Front
  • Back
conjecture
an educated guess based on known information
counterexample
an example that shows that a conjecture is not true.
statement
any statement that is either true, or false, but not both.
truth value
the truth or falsity of a statement
negation of a statement
a statement that has the opposite meaning as well as an opposite truth value.

~p read "not p"
compound statement
joins two or more statements
conjunction
a compound statement formed by joining two or more statements with the word "and"

p ⋀ q read "p and q"

a conjunction is true only when both statements are true.
disjunction
a compound statement formed by joining two or more statements with the word "or"

p ⋁ q read "p and q"

a disjunction is true when at least one of the statements are true.
truth table
a way to organize truth values of statements
conditional statement
a statement that can be written in "if-then" form

p → q read "if p, then q" or "p imples q"
hypothesis
phrase immediately following the word "if"
conclusion
phrase immediately following the word "then"
conditional statement
p → q
converse statement
q → p
inverse statement
~p → ~q
contrapositive statement
~q → ~p
logically equivalent statement
conditional statement with the same truth values.
biconditional statement
the conjunction of a conditional and its converse.

(p → q) ⋀ (q → p) = p ↔ q read: "if and only if"
inductive reasoning
reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction
deductive reasoning
the process that uses facts, rules, definitions, or properties to reach logical conclusions.
Law of Detachment
If p → q is true and p is true, then q is true.

Symbols: [(p → q) ∧ p] → q
Law of Syllogism
If p → q and q → r are true, then p → r is true.

Symbols: [(p → q) ∧ (q → r)] → (p → r)
Reflexive Property
For every real number a, a = a.
Symmetric Property
For all real numbers a and b, if a = b,
then b = a.
Transitive Property
For all real numbers a, b, and c, if a = b and b = c, then a = c.
Addition Property
For all real numbers a, b, c, if a = b,
then a + c = b + c.
Subtraction Property
For all real numbers a, b, c, if a = b,
then a - c = b - c.
Multiplication Property
For all real numbers a, b, and c, ac = bc.
Division Property
For all real numbers a, b, and c, if a = b and c ≠ 0,
then a ÷ c = b ÷ c.
Substitution Property
For all real numbers a and b, if a = b then a may be replaced by b in any equation or expression.
Distributive Property
For all real numbers a, b, and c,
a(b + c) = ab + ac.

"Rainbow Rule"
Commutative Property
For all real numbers a and b,
a + b = b + a (for addition), and
ab = ba (for multiplication)
Associative Property
For all real numbers a, b, and c,
a + (b + c) = (a + b) + c (for addition), and
a(bc) = (ab)c (for multiplication)
deductive proof
a group of algebraic steps used to solve a problem
formal proof
or
two column proof
statements and reasons justifying each statement organized in two columns
postulate
or
axiom
a statement that describes fundamental relationship between the basic terms of geometry
line
Through any two points, there is exactly one line.

A line contains at least two points.
plane
Through any three points not on the same line, there is exactly one plane.

A plane contains at least three points not on the same line.
If two points lie in a plane, then...
...the entire line containing those points lies in that plane.
If two lines intersect, then...
...their intersection is exactly one point.
If two planes intersect, then...
...their intersection is a line.
theorem
a statement or conjecture that has shown to be true
proof
a logical argument in which each statement is supported by statement that is accepted as true
Midpoint Theorem
If M is the midpoint of AB,
then AM ≅ MB.
Ruler Postulate
The points on any line or line segment can be paired with real numbers so that, given any two points A and B on a line, A corresponds to zero, and B corresponds to a positive real number.
Segment Addition Postulate
If B is between A and C,
then AB + BC = AC.
Protractor Postulate
Given AB and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on either side of AB, such that the measure of the angle formed is r.
Angle Addition Postulate
R is in the interior of ∠PQS, if and only if m∠PQR + m∠RQS = m∠PQS.
Supplement Theorem
If two angles form a linear pair, then they are supplementary.
Complement Theorem
If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.
Angles supplementary to the same angle or to congruent angles are ...
... congruent.
Angles complementary to the same angle or to congruent angles are...
... congruent.
Vertical Angles Theorem
If two angles are vertical angles, then they are congruent.
Right Angles
Perpendicular lines intersect to form four right angles.

All right angles are congruent.

Perpendicular lines form congruent adjacent angles.

If two angles are congruent and supplementary, then each angle is a right angle.

If two congruent angles for a linear pair, then they are right angles.