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Proof:
A proof is a logical argument that establishes the truth of a statement beyond any doubt.

Statement:
A statement is a sentence expressed in words (or mathematical symbols)that is either true or false.

Statements do not include: ?
Statements do not include exclamations, questions, or orders.

A statement is simple when ______
A statement is simple when it cannot be broken down into other statements.

A statement is composite when ______
A statement is composite when it is built by using several simple statements connected by punctuation and/or words such as and, although, or, thus, then, therefore, because, for,moreover, however, and so on...

Tautology:
A statement that is always true (e.g.,“A white horse is white.” “Either you have a dollar bill or you do not.”) is called a tautology.

Paradox:
A sentence whose truth cannot be established is called a paradox.

Example of a Paradox:
“This sentence is false.”If we decide that the sentence is true,then it is indeed true that the sentence is false! If we decide that the sentence is false, then it is false that the sentence is false. Thus the sentence must be true!

Hypothesis:
A statement that it is assumed to be true, and from which some consequence follows.(e.g., In the sentence: “If we work on this problem, we will understand it better” the statement “we work on this problem” is the hypothesis.)

In the sentence: “If we work on this problem, we will understand it better” the statement “we work on this problem” is the _______.
In the sentence: “If we work on this problem, we will understand it better” the statement “we work on this problem” is the hypothesis.

Conclusion:
A conclusion is a statement that follows as a consequence from previously assumed conditions (hypotheses).

In mathematics, the ______ is the ‘closing’ of a logical argument.
In mathematics, the conclusion is the ‘closing’ of a logical argument.

Definition:
Definition: A definition is an unequivocal statement of the precise meaning of a word or phrase, a mathematical symbol or concept,to end all possible confusion.

Theorem:
A theorem is a mathematical statement whose truth can be established using logical reasoning on the basis of certain assumptions that are explicitly given or implied in the statement (i.e.,by constructing a proof).

Lemma:
A lemma is an auxiliary theorem proved beforehand, so that it can be used in the proof of another theorem.

Corollary:
A corollary is a theorem that follows logically and easily from a theorem already proved.

The James & James Mathematics Dictionary defines a corollary as a ________.
The James & James Mathematics Dictionary defines a corollary as a “byproduct of another theorem.”

The first step, whether we are trying to prove a result on our own or we are trying to understand someone else’s proof, consists of _________.
clearly understanding the assumptions (hypotheses)made in the statement of the theorem and the conclusion to be reached.

An argument is valid if __?___
An argument is valid if its hypothesis supplies a sufficient and certain basis for the conclusion to be reached.

An argument can be valid and false if ___
at least one of the hypotheses is false.

Example of an argument being valid and false:
All birds are able to fly.Penguins are birds. Therefore penguins are able to fly.

Example of an argument being invalid and reaching a true conclusion.
Cows have four legs. Giraffes have four legs. Therefore giraffes are taller than cows.

