Deductive Reasoning2-3

Deductive Reasoning

The process of reasoning logically from a given statement to a conclusion.

Law of detachment – if a conditional’s hypothesis is true, then its conclusion is true

If and p is true then so is q.

Example 1

Suppose that a mechanic knows that if a car has a dead battery, the car will not start. A mechanic begins to work on a car and finds the battery is dead. What conclusion can she make?

Example 2

Given: If M is the midpoint of a segment, then it divides the segment into two congruent segments.

M is the midpoint of AB

Example 3

If a baseball player is a pitcher, then that player should not pitch a complete game two days in a row. Vladimir Nunez is a pitcher. On Monday, he pitches a complete game. What can you conclude?

Example 4

Does the following argument illustrate the Law of Detachment?

Given: If it is snowing, then the temperature is less than or equal to 32. The temperature is 20.

You conclude: It must be snowing.

Example 5

If possible use the Law of Detachment to draw a conclusion. If it is not possible explain why.

Given: If a road is icy, then driving conditions are hazardous. Driving conditions are hazardous.

Law of Syllogism

Allows you to make conclusions about two different conditional statement when the conclusion of one is the hypothesis of the other.

If and and p is true, we can use the Law of Syllogism to say that r is also true

Example 6

If a number is prime, then it does not have any repeated factors.

If a number does not have any repeated factors, then it is not prime.

Draw a conclusion using the law of syllogism.

Example 7

If a number ends in 0 it is divisible by 10 If a number is divisible by 10, then it is divisible

by 5.

If possible, use the Law of Syllogism to draw a conclusion.

Example 8

If a number ends in 6, then it is divisible by 2. If a number ends is 4, then it is divisible by 2.

If possible use the Law of Syllogism to to make a conclusion.

Homework

Pg 85 #1-21, 27-32