Topic 2 Deductive Reasoning Unit 1 Topic 2

Explore Deduction is a process of reasoning from statements accepted as true to a conclusion. For example, Statement 1: Edmonton is in Alberta. Statement 2: Alberta is in Canada. Conclusion: Therefore, Edmonton is in Canada. Deduction is the process of using logic to show that, if the statements are true, the conclusion must necessarily be true.

Explore Consider each of the following. Decide if deductive reasoning was appropriately applied. 1. All mammals are warm blooded. A dog is a mammal. Therefore, a dog is warm blooded. 2. Some students like pizza. Tony is student. Therefore, Tony likes pizza. 3. Two rectangles have the same area. Therefore, their sides have the same length. 4. Two squares have the same area. Therefore, their sides have the same length. Try this on your own first!!!!

Explore Consider each of the following. Decide if deductive reasoning was appropriately applied. 1. All mammals are warm blooded. A dog is a mammal. Therefore, a dog is warm blooded. Yes, deductive reasoning was appropriately applied. 2. Some students like pizza. Tony is student. Therefore, Tony likes pizza. No, deductive reasoning was appropriately applied. 3. Two rectangles have the same area. Therefore, their sides have the same length. No, deductive reasoning was appropriately applied. 4. Two squares have the same area. Therefore, their sides have the same length. Yes, deductive reasoning was appropriately applied. 8 m 2 m 16 m 4 m A=LW A=32 m 2

Information Inductive reasoning is where the general conclusion is developed by observing patterns and identifying properties in specific examples. Deductive reasoning is where a statement or conclusion is developed based on true statements.

Example 1 Writing Conclusions Write a valid conclusion which can be deduced from the statements. a)Every whole number is an integer. Six is a whole number. b)Water freezes below 0˚C. The temperature is -15˚C. c)All quadrilaterals have four vertices. A parallelogram is a quadrilateral. d)All dogs are mammals. All mammals are vertebrates. Shaggy is a dog. e)All runners train on a daily basis. Jen is a runner. Try this on your own first!!!!

Example 1: Solution a)Every whole number is an integer. Six is a whole number. b)Water freezes below 0˚C. The temperature is -15˚C. c)All quadrilaterals have four vertices. A parallelogram is a quadrilateral. d)All dogs are mammals. All mammals are vertebrates. Shaggy is a dog. e)All runners train on a daily basis. Jen is a runner. Therefore six is an integer  Therefore t he water is frozen. Therefore a parallelogram has four vertices Therefore S haggy is a mammal.  Therefore Jen trains on a daily basis.

Example 2 Contrasting Inductive and Deductive Reasoning Emery claimed that the sum of two consecutive numbers always equals 2 times the first number plus 1. a) Use inductive reasoning to find an example that supports Emery’s conjecture. b) A proof is a mathematical argument showing that a statement is valid in all cases, or that no counterexample exists. Algebra can be used to deductively prove a conclusion. Use algebra to prove Emery’s conjecture. Try this on your own first!!!!

Example 2 Contrasting Inductive and Deductive Reasoning Emery claimed that the sum of two consecutive numbers always equals 2 times the first number plus 1. a) Use inductive reasoning to find an example that supports Emery’s conjecture. b) A proof is a mathematical argument showing that a statement is valid in all cases, or that no counterexample exists. Algebra can be used to deductively prove a conclusion. Use algebra to prove Emery’s conjecture. Try this on your own first!!!!

Example 3 Proving by Deductive Reasoning Use deductive reasoning to prove the following statements. a)The sum of a number and 6 more than that number results in an even answer. b)When five consecutive integers are added together, the sum is always 5 times the median of the numbers. c)A number taken away from 5 times that number results in a multiple of 4. Try this on your own first!!!!

Example 3: Solution a)The sum of a number and 6 more than that number results in an even answer. Since 2x+6 can be divided by 2, the expression results in an even answer.

Example 3: Solution b)When five consecutive integers are added together, the sum is always 5 times the median of the numbers. The median number is (x+2), so you can multiply the median by 5 to show that the two solutions are the same.

Example 3: Solution c)A number taken away from 5 times that number results in a multiple of 4. Since 4x can be divided by 4, the expression results in a multiple of 4.

Example 4 General Cases Try this on your own first!!!! a)Complete the following table. Start by choosing any two numbers, labelled in the table as Value 1 and Value 2. Value 1Value 2 Choose a number. Double it. Add 5. Add the original number. Add 7. Divide by 3. Subtract the original number. b) State a conjecture about the final result.

Example 4: Solution a)Choose any values, first. I choose 6 and 7. Then, complete the table, step by step for each number. Value 1Value 2 Choose a number. 67 Double it Add Add the original number Add Divide by Subtract the original number. 44 b) Conjecture: When you follow the steps, the answer will be 4.

Example 4 c) Using deductive reasoning, prove your conjecture for the general case. General Case Choose a number. Double it. Add 5. Add the original number. Add 7. Divide by 3. Subtract the original number. Try this on your own first!!!!

Example 4: Solution c) To solve this deductively, complete the same chart, but use letters to represent the numbers to prove the general case. General Case Choose a number.x Double it.2x Add 5.2x+5 Add the original number.2x+5+x=3x+5 Add 7.3x+5+7=3x+12 Divide by 3.(3x+12)/3=x+4 Subtract the original number.x+4-x = 4

Need to Know: Deductive reasoning starts with general theories that are true and, through logical reasoning, arrives at a specific conclusion. A conjecture has been proven only when it has been proved true for every possible case or example. This is done through general cases. You’re ready! Try the homework from this section.