9.10.10 1.7 and 1.8

1.7- Deductive Structure

1.8- Statements of Logic

The deductive structure contains: undefined terms, postulates, definitions, and conclusions



Postulate (or Axiom) - a statement that is accepted as true without proof



Theorem- a statement that must be proven true



Undefined terms are to Definitions as Postulates are to Theorems



Conditional Statements:


True vs. False- Truth value

Example:

Sky is blue- True

Grass is purple- False



If (insert hypothesis here), Then (insert conclusion here)

If P, Then Q

Notation: P→Q



P→Q implies that the statement is always true



Example 1:


If you live in Birmingham, then you live in Michigan


Controverse: Switches Q and P: Q→P


Example 2:


If you live in Michigan, you live in Birmingham


Inverse: If NOT P, then NOT Q. Opposites

~P→~Q


Example 3:

If you don’t live in Birmingham, then you don’t live in Michigan.

Truth value: sometimes true



Contrapositive: Inverse converse. Switches P and Q and makes them opposite.

                         ~Q→~P

                        The conditional and the contrapositive are both logically equivalent


Example 4:

If you don’t live in Michigan, you don’t live in Birmingham


Arguments: String of statements together. Has no truth value.




Example 5:



Premise 1: If you live in Birmingham, then you live in Michigan  (true)

Premise 2: Elizabeth live in Birmingham                                     (true)

Conclusion: Therefore, Elizabeth lives in Michigan                     (Must be true)

Notation: ∴ Elizabeth lives in Michigan


P→Q     (P implies Q)

P           (is true)

∴Q        (therefore Q must be true)


Chain of Reasoning




P→Q

Q→R

∴P→R

P→Q→R



P→Q→R→S→T

∴P→T

Sorry for the lack of the venn diagrams.  I don't have a program that suppots that on my computer. Sorry!
So anyways, this is what we learned on Friday  Paired with assignment 3.
Blessings, Em J