Week’s Schedule • Mon: Lesson 1.1 Logic • Tue: Lesson 1.2 Patterns • Wed: Lesson 1.3 Conditional Statements • Thu: Lesson 1.3 (continued) conditional statements in symbolic form • Fri: truth tables
Monday' Schedule • Warm-ups • Quiz • Logic lesson • Logic assignment
Introduction • A farmer has a fox, goose and a bag of grain, and one boat to cross a stream, which is only big enough to take one of the three across with him at a time. If left alone together, the fox would eat the goose and the goose would eat the grain. How can the farmer get all three across the stream? • Simpsons clip (start about 4 min into episode)
Logic • Read all the information carefully and completely. • Decide what the question is asking. • Organize the important information. • Use pictures, tables, grids, etc. to help solve the problem. • Think creatively • Does your answer make sense?
Inductive vs Deductive reasoning • Deductive reasoning: Uses facts, definitions, and accepted properties to write a logical argument. • Inductive reasoning: Uses previous examples and patterns to make a conjecture.
ExamplesInductive or Deductive? • Andrea knows that Todd is older than Chan. She also knows that Chan is older than Robin. Andrea reasons deductively that Todd is older than Robin based on accepted statements. • Andrea knows that Robin is a sophomore and Todd is a junior. All the other juniors that Andrea knows are older than Robin. Therefore, Andrea reasons inductively that Todd is older than Robin based on past observations.
Practice • Robert is shopping in a large department store with many floors. He enters the store on the middle floor from a skyway, and immediately goes to the credit department. After making sure his credit is good, he goes up three floors to the housewares department. Then he goes down five floors to the children’s department. Then he goes up six floors to the TV department. Finally, he goes down ten floors to the main entrance of the store, which is on the first floor, and leaves to go to another store down the street. How many floors does the department store have?
Practice • An explorer wishes to cross a barren desert that requires 6 days to cross, but one man can only carry enough food for 4 days. What is the fewest number of other men required to help carry enough food for him to cross?
Tuesday • Warm-ups • Correct Assignment 1.1 logic • Lesson 1.2 patterns • Assignment
Think about… • A man starts a chain letter. He sends the letter to two people and asks each of them to send copies to two additional people. These recipients in turn are asked to send copies to two additional people each. Assuming no duplication, how many people will have received copies of the letter after the twentieth mailing? What pattern was being formed with the mailings?
Predict the next number in the sequence. What is the pattern? 1, 4, 16, 64, . . . 256 (multiplied by 4) –5, -2, 4, 13, . . . 25 (+3, +6, +9, +12) 1, 1, 2, 3, 5, 8, . . . 13 (add previous two to get the next) 1, 2, 4, 7, 11, 16, 22, . . . 29 (+1, +2, +3, +4, etc)
Brain Buster! In order to keep the spectators out of the line of flight, the Air Force arranged the seats for an air show in a “V” shape. Kevin, who loves airplanes, arrived very early and was given the front seat. There were three seats in the second row, and those were filled very quickly. The third row had five seats, which were given to the next five people who came. The following row had seven seats; in fact, this pattern continued all the way back, each row having two more seats than the previous row. The first twenty rows were filled. How many people attended the air show?
Wednesday • Warm-ups • Correct 1.2—Patterns • Lesson 1.3 Conditional Statements • Assignment: 1.3—Conditional Statements
Conditional Statements • A conditional statement is any statement that is written, or can be written, in the if-thenform. • This is a logical statement that contains two parts: • Hypothesis • Conclusion • If today is Wednesday, then tomorrow is Thursday.
Converse • The converseof a conditional statement is formed by switching the hypothesis and conclusion. If today is Wednesday, then tomorrow is Thursday. If tomorrow is Thursday, then today is Wednesday.
Negation • The negation is the opposite of the original statement. • Make the statement negative of what it was. • Use phrases like • Not, no, un, never, can’t, will not, nor, wouldn’t… Today is Tuesday. Today is not Tuesday. There exists a dog that is not brown All dogs are brown.
Inverse • The inverse is found by negating the hypothesis and the conclusion. • Notice the order remains the same! If today is Wednesday, then tomorrow is Thursday. If today is not Wednesday, then tomorrow is not Thursday.
Contrapositive • The contrapositive is formed by switching the order and making both negative. If today is Wednesday, then tomorrow is Thursday. then tomorrow is not Thursday. If today is not Wednesday, Iftomorrow is not Thursday, then today is not Wednesday.
HINT: Remember that the contrapositive (a big long word) is really the combining together of the strategies of two other words: converse and inverse.
Write the statements in if-then form. • 1) Today is Monday. Tomorrow is Tuesday. If today is Monday, then tomorrow is Tuesday. • 2) Today is sunny. It is warm outside. If today is sunny, then it is warm outside. • 3) It is snowing outside. It is cold. If is is snowing outside, then it is cold
Write the negation of the following statements. • 1) It is sunny outside. • It is not sunny outside. • 2) I am not happy. • I am happy. • 3) All birds can fly. There exists a bird that cannot fly.
Write the inverse, converse and contrapositive of the conditional statement. • Conditional statement: If you get a 60% in the class, then you will pass. • Inverse: • Converse: • Contrapositive:
Write the inverse, converse and contrapositive of the conditional statement. • Conditional statement: If you get a 60% in the class, then you will pass. • Inverse: If you do not get a 60% in class, then you will not pass. • Converse: • Contrapositive:
Write the inverse, converse and contrapositive of the conditional statement. • Conditional statement: If you get a 60% in the class, then you will pass. • Inverse: If you do not get a 60% in class, then you will not pass. • Converse: If you pass, then you got a 60% in class. • Contrapositive:
Write the inverse, converse and contrapositive of the conditional statement. • Conditional statement: If you get a 60% in the class, then you will pass. • Inverse: If you do not get a 60% in class, then you will not pass. • Converse: If you pass, then you got a 60% in class. • Contrapositive: If you do not pass, then you did not get a 60% in class.
Equivalent statements • If the conditional statement is true, then the contrapositive statement is also true. Therefore, they are equivalent statements. • If the inverse statement is true, then the converse statement is also true. Therefore, they are equivalent statements.
Biconditional Statement • A biconditional statement is a statement that is written, or can be written, with the phrase if and only if. • If and only if can be written shorthand by iff. • Writing a biconditional is equivalent to writing a conditional and its converse.
Write the following conditional statements as biconditional statements. • 1) If the ceiling fan runs, then the light switch is on. • The ceiling fan runs if and only if the light switch is on. • 2) If you scored a touchdown, then the ball crossed the goal line. You scored a touchdown if and only if the ball crossed the goal line. 3) If the heat is on, then it is cold outside. The heat is on iff it is cold outside.
Thursday • Warm-ups • Correct lesson 1.3—conditional statements • Continue lesson 1.3—conditional statement written in symbolic form • Assignment 1.3
Symbolic Conditional Statements • To represent the hypothesis symbolically, we use the letter p. • We are applying algebra to logic by representing entire phrases using the letter p. • To represent the conclusion, we use the letter q. • To represent the phrase if…then, we use an arrow, . • To represent the phrase if and only if, we use a two headed arrow, .
Example of Symbolic Representation • If today is Tuesday, then tomorrow is Wednesday. • p = Today is Tuesday • q = Tomorrow is Wednesday • Symbolic form p q • We read it to say “If p then q.”
Negation • Recall that negation makes the statement “negative.” • That is done by inserting the words not, nor, or, neither, etc. • The symbol is much like a negative sign but slightly altered… ~
Symbolic Variations • Converse q p • Inverse ~p ~q • Contrapositive ~q ~p • Biconditional p q
Use the statements to construct the propositions. • p: It is a snake. • q: It has scales. 1) 2) 3)
Use the statements to construct the propositions. • p: It is a snake. • q: It has scales. 1) If it is a snake, then it has scales. 2) 3)
Use the statements to construct the propositions. • p: It is a snake. • q: It has scales. 1) If it is a snake, then it has scales. 2) It is not a snake 3)
Use the statements to construct the propositions. • p: It is a snake. • q: It has scales. 1) If it is a snake, then it has scales. 2) It is not a snake 3) If it is not a snake, then it does not have scales.
Law of Detachment • If pq is a true conditional statement and p is true, then q is true. • It should be stated to you that pq is true. • Then it will describe that p happened. • So you can assume that q is going to happen also. • This law is best recognized when you are told that the hypothesis of the conditional statement happened.
Example 2 • If you get a D- or above in Geometry, then you will get credit for the class. • Your final grade is a D. • Therefore… • You will get credit for this class!
Law of Syllogism • If pq and qr are true conditional statements, then pr is true. • This is like combining two conditional statements into one conditional statement. • The new conditional statement is found by taking the hypothesis of the first conditional and using the conclusion of the second. • This law is best recognized when multiple conditional statements are given to you and they share alike phrases.
Example 3 • If tomorrow is Wednesday, then the day after is Thursday. • If the day after is Thursday, then there is a quiz on Thursday. • Therefore… • And this gets phrased using another conditional statement • If tomorrow is Wednesday, then there is a quiz on Thursday.
Are the following logical arguments? If so do they use the law of syllogism or detachment? • Scott knows that if he misses football practice the day before the game, then he will not be a starting player in the game. Scott misses practice on Thursday so he concludes that he will not be able to start in Friday’s game. • If it is Friday, then I am going to the movies. If I go to the movies, then I will get popcorn. Since today is Friday, then I will get popcorn. • If it is Thanksgiving, then I will eat too much. If I eat too much, then I will get sick. I got sick so it must be Thanksgiving. Law of Detachment Law of Syllogism Not a valid argument
Counterexamples • To find a counterexample, use the following method: • Assume that the hypothesis is TRUE. • Find any example that would make the conclusionFALSE.
Find a counterexample • If it can be driven, then it has four wheels. • All boats float. • If it is a bird, then it can fly.
Week’s Schedule • Mon: Truth tables • Tue: Logical/Statistical and Necessary/sufficient • Wed: Algebra proofs • Thu: Review/practice test • Fri: Test
Monday • Warm-ups • Return quizzes • Correct assignment 1.3 Symbolic Notation • Lesson 1.4 truth tables • Assignment • grades
Consider the statement: • “If you clean your room, then your parents will give you $20. • If you clean your room and your parents gave you $20, would you be happy? • If you clean your room and your parents did not give you $20, would that be ok? • If you did not clean your room but your parents gave you $20 anyway, would you be happy? • If you did not clean your room and you did not get the $20, would that be ok?
Truth tables • A truth table displays the relationships between the truth values of propositions. Truth tables are especially useful in determining the truth values of propositions constructed from simpler propositions.