GeometryName: ______
Class Notes
Section 2.3Date: ______Period:______
Deductive Reasoning
Deductive Reasoning: Proving statements by reasoning from accepted postulates, definitions, theorems, and given information.
Inductive Reasoning:
If you like the beach, then you should visit Long BeachIsland.
The conditional statement can be written symbolically as follows:
If p,then qor
To form the converse, switch p and q. The converse can be written symbolically as follows:
If q, then por
A biconditional statement can be written using symbolic notation as follows:
If p, thenq and if q, then p.orpqorp iff q (this is not a typo)
To write the inverse and the contrapositive, you need to know the symbol for negate or not.
represents the negation of p, not p
An inverse statement can be written using symbolic notation as follows:
If , then ~ qor
A contrapositive statement can be written using symbolic notation as follows:
If ~ q then ~ por
Example 1:Let p be “the value of x is -4.” and q be “the square of x is 16.”
- Write in words.
- Write in words.
c.Decide whether the bicondidional statement pqis true.
Example 2:Let p be “today is Monday” and q be “there is school.
a.Write the contrapositive of in symbolic notation and then in words.
b.Write the inverse of in symbolic notation and then in words.
The Law of Detachment
If is a true conditional statement and p is true, then q is true.
The following statement is true, “If today is Thursday, then Mr.Murphy will go running.”
Today is Thursday.
Since it is a Thursday, you can assume Mr. Murphy will go running.
Example 4:If a vehicle is a car, then it has four wheels is a true conditional.
A sedan is a car. Use the Law of Detachment to reach a valid conclusion.
Example 5:Determine if a valid conclusion can be reached from the two true statements,
If I watch TV, I will not do my homework and I did my homework, using the Law of Detachment.
If a valid conclusion is possible, state it.
The Law of Syllogism
If and are true conditional statements, then is true.
The following two statements are true,
“If the Eagles win, then the Cowboys stink.”
“If the Cowboys stink, then I am happy.”
Syllogism allows you to conclude, “If the Eagles win, then I will be happy.”
**Remember in order to apply The Law of Syllogism, two true if-then statements must be given.
Example 6:Determine if a valid conclusion can be reached from the two true statements,
If Jamie takes the car to the store, she will stop at the post office and If Jamie stops at the post office, she will buy stamps, using the Law of Syllogism. If a valid conclusion is possible, state it.
TRY THESE
Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If is does not, write invalid.
#1(1) If you plan to attend the University of Notre Dame, then you need to be in the top 10% of
your class.
(2) Jimmy Moon plans to attend Notre Dame.
(3) Jimmy Moon needs to be in the top 10% of his class.
#2(1) Vertical angles are congruent.
(2)
(3) are vertical.
Determine if a valid conclusion can be reached from the two true statements using the Law of Detachment or the Law of Syllogism. If a valid conclusion is possible, state it and the law used. If a valid conclusion does not follow, write no conclusion.
#3(1) If the measure of an angle is between 0oand 90°, then it is acute.
(2) If an angle is acute, then it is not obtuse.
#4(1) If you want good health, then you should get 8 hours of sleep each night.
(2) Savannah wants good health.
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