Download Laws of Deductive Reasoning

Mrs. Blanco
Geometry
Objectives:
 Use symbolic notation to represent logical
statements.

Form conclusions by applying the laws of logic
to true statements.




Conditional statement
If p, then q
OR p → q
Converse
If q, then p
OR q → p
Inverse
If ~p, then ~ q
OR ~p → ~q
Contrapositive
If ~q, then ~ p
OR ~ q → ~ p

Written symbolically:
If p, then q and if q, then p. or
pq
Most often written in this form:
p if and only if q.
 Let p be “the value of x is -5” and let q be
“the absolute value of x is 5.”
 A. Write p → q in words.
 B. Write ~p → ~ q in words.
 C. Write q → p in words.
 D. Write ~ q → ~ p in words.
 E. Decide whether the biconditional statement
p  q is true.
A. If the value of x is -5, then the absolute value of x is 5.
B. If the value of x is not -5, then the absolute value of x is
not 5.
 C. If the absolute value of x is 5, then the value of x is -5.
 D. If the absolute value of x is not 5, then the value of x is
not -5.
 E. The conditional statement in part a is true, but its
converse (b) is false. So, the biconditional p  q is false.


The Conditional and
Contrapositive are equivalent
statements.
The Converse and Inverse are
equivalent statements.

Definition:
 Deductive reasoning--uses facts,
definitions, and accepted properties in a
logical order to write a logical argument
 Inductive reasoning—uses examples and
patterns are used to form a conjecture.

Andrea knows that Robin is a sophomore and
Todd is a junior. All the other juniors that
Andrea knows are older than Robin.
Therefore, Andrea reasons inductively that
Todd is older than Robin based on past
observations.
1. Law of Detachment
2. Law of Syllogism



If a statement p→q is given and a second
statement p is given, then a third statement q
results.
Given: p→q
p
q
q
p
Ex: 1. If x is even, then x2 is even.
2. x = 6
What statement follows?
62 is even
Example:
If two angles form a linear pair, then they are
supplementary;
A and B form a linear pair.
So,
A and B are supplementary.
Example:
If two angles form a linear pair, then they are
supplementary;
A and B are supplementary.
So,
No Conclusion
 If p→q is given and q→r is given,
then p→r results.
 Given: p→q
q→r
p→r
Given:
p
1. If a figure is a square, then it is a
quadrilateral. q

r
q
2. If a figure is a quadrilateral, then it is a
polygon.
p→q
q→r
p→r
What statement follows?
p
r
If a figure is a square, then it is a polygon.