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### Lesson 2.3 Mathematical Reasoning and Logic

• 1.
• 2. Learning Outcomes: At the end of this lesson, the students will be able to: 1. Analyze problems using different types of reasoning. 2. Apply different types of reasoning to justify statements and arguments made about mathematics and mathematical concepts.
• 3.
• 4. INDUCTIVE REASONING It is the process of reaching a general conclusion by examining specific examples.
• 5. INDUCTIVE REASONING The conclusion formed by using inductive reasoning is often called a conjecture, since it may or may not be correct.
• 6. Example 1: Use inductive reasoning to predict the next number in each item. 1. 2,8,14,20,26, _____ 2. 1,2,5,10,17,26, _____
• 7. Example 2: A. Every sports car I have ever seen is red. Thus, all sports cars are red. B. The coin I pulled from the bag is a 5-peso coin. Another 5-peso coin is drawn from the bag. A third coin from the bag is again a 5-peso coin. Therefore, all the coins in the bag are 5-peso coins.
• 8. Example 3: Consider the following. Pick a number. Multiply the number by 4, add 8 to the product, divide the sum by 2, and subtract 5. Complete the above procedure for several different numbers. Use inductive reasoning to make a conjecture about the relationship between the size of the resulting number and the size of the original number.
• 9. Solution: Suppose we start with seven as the original number. Then repeat the process for different numbers. The procedure yields the following: We conjecture that the given procedure produces a number that is one less than twice the original number.
• 10. Remarks: When we use inductive reasoning, we have no guarantee that our conclusion is correct. Just because a pattern is true for a few cases, it does not mean the pattern will continue. A statement is a true statement provided that it is valid in all cases. If we can find one case for which a statement is not valid, called a counterexample, then it is a false statement.
• 11. DEDUCTIVE REASONING It is the process of reaching a conclusion by applying general assumptions, procedures, or principles.
• 12. DEDUCTIVE REASONING Deduction starts out with a general statement, or hypothesis, and examines the possibilities to reach a specific, logical conclusion.
• 13. Example 1: 1. All men are mortal. Kahwi is a man. Therefore, Kahwi is mortal. 2. Corresponding parts of congruent triangles are congruent. Triangle ABC is congruent to triangle DEF. Angle B and angle E are corresponding angles. Thus, angle B is congruent to angle E.
• 15. Determine if each of the following statement uses inductive or deductive reasoning. 1. Teacher Erica is an enthusiastic and passionate teacher. Therefore, all teachers are enthusiastic and passionate. 2. All dogs are animals. Dhai is a dog. Thus, Dhai is an animal. 3. I got low score on the first long exam. I just recently took the second long exam and I got low score. Therefore, I will also get a low score on the third long exam. IR DR IR
• 16. Determine if each of the following statement uses inductive or deductive reasoning. 4. My classmates are disrespectful toward our instructor. Hence, all students are disrespectful. 5. Last Wednesday it was raining. Today is Wednesday and it is raining. Therefore, on the next Wednesday, it will also rain. 6. For any right triangle, the Pythagorean Theorem holds. ABC is a right triangle, therefore for ABC the Pythagorean Theorem holds. 7. All basketball players in your school are tall, so all basketball players must be tall. IR IR DR IR
• 17. A logic puzzle is a puzzle deriving from the mathematics field of deduction. Logic puzzles can be solved by using deductive reasoning and by organizing the data in a given situation.
• 18. A logic puzzle is basically a description of an event or any situation. Using the clues provided, one has to piece together what actually happened. This involves clear and logical thinking, hence the term “logic” puzzles.
• 19. Example 1 Three musicians appeared at a concert. Their last names were Benton, Lanier, and Rosario. Each plays only one of the following instruments: guitar, piano, or saxophone. 1. Benton and the guitar player arrived at the concert together. 2. The saxophone player performed before Benton. 3. Rosario wished the guitar player good luck. Who played each instrument? Guitar Piano Saxophone Benton Lanier Rosario
• 21. Example 3 A census-taker knocks on a door, and asks the woman inside how many children she has and how old they are. “I have three daughters, their ages are whole numbers, and the product of their ages is 36,” says the mother. “That’s not enough information”, responds the census-taker. “I’d tell you the sum of their ages, but you’d still be stumped.” “I wish you’d tell me something more.” “Okay, my oldest daughter Annie likes dogs.” What are the ages of the three daughters (Zeitz, 2007)?
• 22. Solution: After the first reading, it seems impossible- there isn’t enough information to determine the ages. The product of the ages is 36, so there are only a few possible triples of ages. Here is a table of all the possibilities Age 1, 1,36 1, 2,18 1, 3, 12 1, 4, 9 1, 6, 6 2, 2, 9 2, 3, 6 3, 3, 4 Sum 38 21 16 14 13 13 11 10 Now we see what is going on. The mother’s second statement (“I’d tell you the sum of their ages, but you’d still be stumped.) gives valuable information. It tells that the ages are either 1,6,6 or 2,2,9, for in all other cases, knowledge of the sum would tell unambiguously what the ages are. The final clue now makes sense, it tells that there is an oldest daughter, eliminating 1, 6,6. The daughters are thus 2, 2, and 9 years old.

### Editor's Notes

1. Reasoning is a process based on experience and principles that allow one to arrive at a conclusion.
2. Reasoning is a process based on experience and principles that allow one to arrive at a conclusion.
3. We make generalizations from the part to the whole. If we are not careful, it can lead to erroneous or mistake conclusions:
4. Conjecture- estimation or guess
5. Each successive number is 6 larger than the preceding number.
6. It is not enough that the deduction is logically sound; the assumption (1) must also be true. Consider the following: "1) All cats are red. 2) Kitty is a cat, therefore Kitty is red." It is logically valid but leads to a non-valid conclusion because not 'all cats are red'.
7. In mathematics, deductive reasoning makes use of definitions, axioms, theorems and rules and inference.
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