Deductive Reasoning Deductive Reasoning Definition Use facts definitions

Deductive Reasoning

Deductive Reasoning Definition: Use facts, definitions, accepted properties & laws of logic to form a logical argument.

Laws of Logic (2) �Law of Detachment- if the hypothesis of a true conditional statement is true, then the conclusion is also true. (also called a direct argument)

Law of Detachment Example �Conditional Statement: If you run every day, then you get good exercise. (general statement-’you’ could mean anyone)

Law of Detachment Example Adding a specific hypothesis to go with conditional statement If you run every day, then you get good exercise. Hypothesis: Emilio runs every day.

What can we conclude? Emilio gets good exercise. We can conclude he gets good exercise by: Law of Detachment

Example 1 -Use Law of Detachment � Conditional Statement: If two angles have the same measure, then they are congruent. You know m∠A = m∠B. Solution Because m∠A = m∠B satisfies the hypothesis of a true conditional statement, the conclusion is true. So, m∠A ≅ m∠B.

Example 1 b. Jesse goes to the gym every weekday. CONCLUSION HYPOTHESIS IDENTIFY THE HYPOTHESIS AND THE CONCLUSION. If it is a weekday, then Jesse goes to the gym.

If it is a weekday, then Jesse goes to the gym. Today is Monday satisfies the hypothesis, so we can conclude that……. . Jesse will go the gym today!

Law of Syllogism � If you have 2 conditional statements & the conclusion of one is the same as the hypothesis of another, then you can conclude: the hypothesis of 1 st one implies the conclusion of the 2 nd. H 1 C 1 H 2 C 2

Example �If you live in Austin, then you live in Texas. �If you live in Texas, then you live in the Lone Star State. Use Law of Syllogism to “simplify” into 1 conditional statement

If you live in Austin, then you live in Texas. If you live in Texas, then you live in the Lone Star State.

Conclusion by Law of Syllogism If you live in Austin, then you live in the Lone Star State.

Example 2 A If Ron eats lunch today, then he will eat a sandwich. If Ron eats a sandwich, then he will drink a glass of milk. Use the Law of Syllogism: If Ron eats lunch today, then he will drink a glass of milk.

Example 2 B If x 2 > 36, then x 2 > 30. If x > 6, then x 2 > 36. If x > 6, then 2 x > 30.

Example 3 C If a triangle is equilateral, then all of its sides are congruent. If a triangle is equilateral, then all angles in the interior are congruent. What can you conclude? ? ?

� Since the conclusion is not the same as the other’s hypothesis, you cannot use the Law of Syllogism.

CHECKPOINT 1. Using the LAW OF DETACHMENT, you can say: ∠A is acute. 2. If you do your homework, then you can watch TV. If you watch TV, then you can watch your favorite show. If you do your homework, then you can watch your favorite show. F O W LA M S I G O L L SY