Geometry Chapter 2.3 Deductive Reasoning
I) Using Symbolic Notation:
1) " → " means implies
2) "~" means not
3) "↔" means if and only if
A) conditional statement has a hypothesis (symbolically "p") and a conclusion (symbolically "q") so: if p then q or p → q
B) Converse: q → p
C) Inverse: ~p → ~ q
D) Contrapositive: ~q → ~ p
E) Biconditional: p ↔ q
F) Deductive Reasoning: uses facts, definitions, and accepted properties in a logical order to write a logical argument.
1) Law of detachment: if "p → q" is a true statement and "p" is a true statement, then we can conclude that "q" is true.
Example: If Mark gets time off of work then he is going to Hawaii on vacation. (this is true)
Mark gets time off of work. (this is true).
Therefore we can conclude that Mark is going to Hawaii on vacation.
Symbolically:
Let "p" be Mark gets time off of work.
Let "q" be Mark is going to Hawaii on vacation.
p → q is true
p is true.
Therefore q is true.
2) Law of Syllogism:
If p → q AND q → r are both true, then we can conclude that p → r .
This is like the transitivity postulate.
Example: If Carol buys a new swimsuit, then she is going to the Lake.
If Carol goes to the Lake, then she will go swimming.
Therefore we can conclude that
If Carol buys a new swimsuit, then she will go swimming.
2.4 Reasoning with Properties from Algebra:
G) Postulates:
1. Reflexive Postulate: a = a
2. Symmetric Postulate: if a = b then b = a
3. Transitivity Postulate: if a = b and b = c, then a = c
4. Addition Postulate: if a = b and c is not equal to 0, then a + c = b + c
5. Subtraction Postulate: if a = b and c is not equal to 0, then a - c = b - c
6. Multiplication Postulate: if a = b and c is not equal to 0, then ac = bc
7. Division Postulate: if a = b and c is not equal to 0, then a/c = b/c
8. Distribution Postulate: if a(b + c) then ab + ac
H) Definitions:
1. Perpendicular lines - form right angles
2. Right angle - an angle that measures 90 degrees.
