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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? 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. Examples Inductive 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? Find the pattern and then predict the next image. 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 ifthen form. – This is a logical statement that contains two parts: • Hypothesis • Conclusion • If today is Wednesday, then tomorrow is Thursday. Converse • The converse of 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. All dogs are brown. Today is not Tuesday. There exists a dog that is not 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. The Inverse Mohawk Contrapositive • The contrapositive is formed by switching the order and making both negative. If today is Wednesday, then tomorrow is Thursday. If today is not Wednesday, then tomorrow is not Thursday. If tomorrow 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 pq • 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 qp • 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) pq 2) p 3) p q Use the statements to construct the propositions. • p: It is a snake. • q: It has scales. 1) pq If it is a snake, then it has scales. 2) p 3) p q Use the statements to construct the propositions. • • 1) p: It is a snake. q: It has scales. pq If it is a snake, then it has scales. 2) p It is not a snake 3) p q Use the statements to construct the propositions. • • 1) p: It is a snake. q: It has scales. pq If it is a snake, then it has scales. 2) p It is not a snake 3) p q 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 • 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 • 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. Law of Detachment • 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. Law of Syllogism • 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. 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 conclusion FALSE. 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. Friday • • • • Warm-ups Correct assignment 1.3 Symbolic Notation Lesson 1.4 truth tables Assignment Consider the statement: • “If today is Friday, then 2 + 3 = 6.” • Is this statement true if it is Wednesday • Is this statement ever true or is it always false? • “If it is sunny today, then we will go to the beach.” • When is this statement true and when is this statement false? 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. Symbols • Review symbols: negation if-then (conditional) iff (biconditional) • New symbols: and or Rules—Let p and q be propositions • “p and q”, denoted by p q , is true when both p and q are true and is false otherwise. Rules —Let p and q be propositions • “p and q”, denoted by p q , is true when both p and q are true and is false otherwise. • “p or q”, denoted by p q , is false when p and q are both false and true otherwise. Rules —Let p and q be propositions • “p and q”, denoted by p q , is true when both p and q are true and is false otherwise. • “p or q”, denoted by p q , is false when p and q are both false and true otherwise. • “If p, then q”, denoted by p q , is false when p is true and q is false and true otherwise. Rules —Let p and q be propositions • “p and q”, denoted by p q , is true when both p and q are true and is false otherwise. • “p or q”, denoted by p q , is false when p and q are both false and true otherwise. • “If p, then q”, denoted by p q , is false when p is true and q is false and true otherwise. • “p if and only if q”, denoted by p q , is true when p and q have the same truth values and is false otherwise. Practice Truth Table for pq p q pq T T T T F F F T F F F F Practice Truth Table for p q p q pq T T T T F F F T T F F T Practice Truth Table for p q p q pq T T T T F T F T T F F F Practice Truth Table for p q p q pq T T T T F F F T F F F T Practice Truth Table for p q p q q p T T F F T F T T F T F F F F T F q Practice Truth Table ( p forq ) ( p q) p q pq pq T T T T T T F T F F F T T F F F F F F T ( p q) ( p q) Practice Truth Table (p q )for ( p q) ( p q) ( p q) p q pq T T T F T T T F F F T T F T T T T T F F T T F F p pq Monday • Warm-ups • Correct 1.4 truth tables • Lesson 1.5 Logical vs Statistical arguments Necessary and Sufficient • Assignment 1.5 Logical or Statistical? What kind of argument will you make? • You are playing a game of Old Maid. In your current hand of five cards, one is the Old Maid. Give an argument explaining the odds that after your opponent draws, you are still stuck holding the Old Maid. Logical vs Statistical • Logical: Based on reason or what is expected. • Statistical: Based on data, examples, experimentation, numerical facts, etc. Make a statistical and a logical argument for the given situation. • You are rolling a number cube (dice) with the numbers 1-6 on it. What is the chance of getting an even number versus an odd. – Statistical: ½ or 50%. (3 even out of 6 total) – Logical: Chances are even because there are the same number as evens as there are odds. Make a statistical and a logical argument for the given situation. • Drawing from a deck that has 10 black cards and 5 red cards, do you think the next card will be red? – Statistical: No, the next card only has a 5 out of 15, or 33%, chance of being red. – Logical: No, there are a twice as many black cards as there are red. Necessary • If we say that "x is a necessary condition for y," we mean that if we don't have x, then we won't have y • To say that x is a necessary condition for y does not mean that x guarantees y. • Water is necessary for plant life. However, it water does not guarantee plant life. Sufficient • If we say that "x is a sufficient condition for y," then we mean that if we have x, we know that y must follow • In other words, x guarantees y. • Rain pouring from the sky is a sufficient condition for the ground to be wet. Necessary or Sufficient? • Earning a total of 95% in class and getting a grade of an A. Sufficient • Having gas in my car and my car to starting. Necessary • Pouring a gallon of freezing water on my sister and her waking up. Sufficient Tuesday Let’s try… • Start out with the statement: 5 = 5 • Add 3 to both sides of the equal sign. What do you notice? • Subtract 3 from both sides of the equal sign. What do you notice? • Divide both sides of the equation by 5. What do you notice? • Multiply both sides of the equation by 3. What do you notice? Vocabulary • Theorem: A true statement that follows as a result of other true statements • Postulate: A rule that is accepted without proof. • Proof: A sequence of justified conclusions used to prove the validity of a statement. • Conjecture: An unproven statement that is based on observations. Algebraic proofs • An algebraic proof basically involves writing a reason for each step while solving an equation. Algebraic properties of equality Property Definition Identification Abbreviation Addition Property If a=b, then a+c = b+c. Something is added to both sides of the equation. APOE Subtraction Property If a=b, then a-c = b-c. Something is subtracted from both sides of the equation. SPOE Multiplication Property If a=b, then ac = bc. Something is multiplied to both sides of the equation. MPOE Division Property If a=b and c≠0, then a/c = b/c. Something is being divided into both sides. DPOE Property Substitution If a=b, then a can be substituted for b in any expression. One object is used in place of another without any calculations being done. SUB Distributive Property a(b+c) = ab + ac A number outside of parentheses has been multiplied to all numbers inside. DIST Example Solve 9x+18=72 9x+18=72 Given 9x=54 SPOE -18 9 -18 9 x=6 DPOE Short for “Information given to us.” Example If 5x+3x-9=79, then x=11 5x+3x-9=79 8x-9=79 9 Given Dist 9 x=6 DPOE