Day 5 Notes 24 Deductive Reasoning
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Deductive reasoning is the process of using facts, rules,definitions, or properties to reach conclusions
Determine whether each conclusion is based on inductive or deductive reasoning. a. Every time Kennedy has skipped studying for a science test, she has gotten at least six wrong. Kennedy skipped studying for her science test today, so she concludes that she will get at least six wrong.
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b. If Kyle is late enrolling for summer school, he will be assessed a late fee of $30. Kyle's enrolled late this summer, so he concludes that he will be assessed a late fee of $30.
* Law of Detachment One form of deductive reasoning that draws conclusions from a true conditional p →q and a true statement p is called the Law of Detachment.
* If p → q is true and p is true, then q is true. Example Given: If a car is out of gas, then it will not start. Sarah's car is out of gas. Valid Conclusion: Sarah's car will not start.
True or False ?
1. All horses are mammals. Mr. Ed is a horse. Therefore, Mr. Ed is a mammal.
2. All horses are mammals. George is a mammal. Therefore, George is a horse.
3. If a family buys a new home in Neighborhood A, they will pay more than $150,000 for the home. The Thomas’ bought a new home in Neighborhood A. Therefore, the Thomas’ pay more than $150,000 for a new home.
4. Some quadratic functions have only one xintercept. Function g has only one xintercept. Therefore, function g is a quadratic function.
Valid. The given statement satisfies the hypothesis, so p is true. By the Law of detachment, the conclusion matches q, is true.
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The given statement satisfies the conclusion q of the true conditional. However, knowing that a conditional statement and its conclusion are true does not make the hypothesis true. The conclusion is invalid.
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Day 5 Notes 24 Deductive Reasoning Law of Syllogism
Another way to make a valid conclusion is to use the Law of Syllogism. It is similar to the Transitive Property.
* If p → q is true and q → r is true, then p → r is also true. Example
Given: If you get a job, then you will earn money. If you earn money, then you will buy a car. Conclusion: If you get a job, then you will buy a car.
Valid or Invalid ? If there is lightning, then it is not safe. If it is not safe, then you should take cover. If you take cover, then it is lightning
If there is lightning, then it is not safe. If it is not safe, then you should take cover. If there is lightning, you should take cover.
Determine whether you can use the Law of Syllogism or the Law of Detachment to reach a valid conclusion from each set of statements. If it does, state which law was used and write the conclusion. If it does not, write invalid. 1. If a number is a whole number, then the number is an integer. * Law of Syllogism If a number is an integer, then it is a rational number.
* If a number is a whole number, then it is a rational number. 2. If an angle is supplementary to an obtuse angle, then it is acute. * Law of Syllogism If an angle is acute, then its measure is less than 90.
* If an angle is supplementary to an obtuse angle, then its measure is less than 90. 3. If a dog eats Super Dog Food, he will be happy. Rover eats Super Dog Food.
* Law of Detachment
* Rover is happy. 4. If an angle is a right angle, then the measure of the angle is 90.* Law of Syllogism If two lines are perpendicular, then they form a right angle.
* If two lines are perpendicular, then the measure of the angle is 90. 5. If you study for the test, then you will receive a high grade. Your grade on the test is high.
* Invalid. No conclusion is possible. 6. If you are a football player then you are an athlete. * Law of Detachment Javier is a foot ball player.
* Javier is an athlete.
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7. If an angle is acute then it measures less than 90. * Law of Detachment
* The measurement of angle A is less than 90.
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