Logic in Mathematics and Deductive Reasoning

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Delve into the principles of logical deduction in mathematics through examples of conditional statements, syllogisms, and proofs. Explore how deductive reasoning can lead to valid conclusions based on given premises.

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WARM UP 1. 1. Write a two column proof for the following problem: Given: ?+5 Prove: x=15 4=5 2. If possible, write the bi-conditional form of the statement below: Palm trees grow in warm climates

ANNOUNCEMENTS Test Thursday! Homework 1.Pg: 62-63 # s 1-4 & 6-11 2.Work on review to study for Test on Thursday

WHAT CAN WE CONCLUDE??? Sherlock Holmes knows that Sherlock Holmes knows that whoever was in the kitchen whoever was in the kitchen stole the tomatoes stole the tomatoes Sherlock Holmes discovers that Sherlock Holmes discovers that Mrs. Hudson was in the kitchen Mrs. Hudson was in the kitchen What can he conclude? What can he conclude?

DEDUCTIVE REASONING The process of using logic to get The process of using logic to get from what you know is true to new from what you know is true to new conclusions conclusions If the statements are true (what you If the statements are true (what you know) what you conclude will also know) what you conclude will also be true be true

TYPE ONE If Mrs. Hendricks is hungry then she is If Mrs. Hendricks is hungry then she is grumpy grumpy Mrs. Hendricks is hungry Mrs. Hendricks is hungry What can we conclude??? What can we conclude???

LAW OF DETACHMENT If a conditional statement its true, and its hypothesis is true, then its conclusion will be true as well. Symbolic form: If p q is a true statement and p is true then q is true ** means if then statement ** means if then statement

TYPE TWO If it rains the flowers will grow If the flowers grow Mrs. Hendricks will be happy THEREFORE If it rains Mrs. Hendricks will be __________ WHAT DID WE DO??? What do you notice about the first two conditional statements?

LAW OF SYLLOGISM Allows you to state one conclusion from 2 true conditional statements when the conclusion from one statement is the hypothesis other statement. conclusion hypothesis from the Symbolic Form: If p q and q r are true statements, then p r is a true statement ** means if then statement

MIND PUZZLES Who killed Colonel Mustard? Who killed Colonel Mustard? Things you know: Things you know: If someone was in the lounge, then they were the perpetrator that killed If someone was in the lounge, then they were the perpetrator that killed Colonel Mustard Colonel Mustard If professor plum was in the kitchen then Miss Peacock was in the If professor plum was in the kitchen then Miss Peacock was in the library library If Ms. Peacock was in the library then Miss Scarlet was in the lounge If Ms. Peacock was in the library then Miss Scarlet was in the lounge If Ms. Peacock was in the lavatory then Mr. White was in the lounge If Ms. Peacock was in the lavatory then Mr. White was in the lounge If Mr. Green was hungry, then Mr. Plum would make him sandwich in If Mr. Green was hungry, then Mr. Plum would make him sandwich in the kitchen the kitchen Mr. Green was hungry Mr. Green was hungry

Who Took The Candy Bar? During the lunch hour at school, a group of five boys from Miss Jones home room visited a nearby lunch wagon. One of the five boys took a candy bar without paying for it. When the boys were questioned by the school principal, they made the following statements in respective order: 1. Rex: "Neither Earl nor I did it." 2. Jack: "It was Rex or Abe." 3. Abe: "Both Rex and Jack are lying." 4. Dan: "Abe's statement is not true; one of them is lying and the other is speaking the truth." 5. Earl: "What Dan said is wrong." When Miss Jones was consulted, she said, "Three of these boys are always truthful, but everything that two of them say will be a lie." Assuming that Miss Jones is correct, can you determine who took the candy bar?