Geometry Page G 7

G 7

COMPONENTS OF A FORMAL AXIOMATIC SYSTEM:

1. Undefined (primitive) technical terms;

2. Unsupported statements (called axioms) about the undefined terms;

3. Technical terms (called definitions) defined by previously introduced

terms;

4. Statements (called theorems) deduced from previous statements.

A FEW NOTES ON LOGIC:

The statement, if p, then q (p -> q), is called a conditional statement or an implication. Its converse is the statement, if q, then p (q ->p).
The truth of an implication does not imply the truth of its converse.
If p -> q is true, and if q-> p is true, then p and q are equivalent.
The implication, if not q, then not p, is the contrapositive of the implication, if p, then q.
An implication and its contrapositive are either both true or both false.
The statement, p if and only if q (p <-> q), is called a biconditional statement.

Theorems are proven using various rules of logic. There are many. Some follow direct reasoning having the following "form":

If p, then q

If q. then z

Thus, If p, then z

Also, proof by contradiction (also called "indirect proof" or "reduction to absurdity") can be used to prove statements. Two rules are involved:

Law of the Excluded Middle: either a statement is true or its negation is true;

Law of Contradiction: a statement and is negation cannot both be true.

Proof by contradiction generally takes the following "form":

To prove, if p, then q, first assume that not q is true. Show that this assumption contradicts some previous component of the system.

In proof, six types of justification are allowed: 1) by hypothesis; 2) by axiom...;

3) by a theorem... (previously proved); 4) by definition ... ; 5) by step ... ; and

6) by rule ... of logic