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 childrens 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?
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?
Conditional Statements A conditional statement is any statement that is written, or can be written, in the if- then 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 tomorrow is Thursday, If today is Wednesday, then 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, cant, will not, nor, wouldnt Today is Tuesday.Today is not Tuesday. All dogs are brown. 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 not Wednesday, If today is Wednesday, then tomorrow is Thursday. then tomorrow is not Thursday.
The Inverse Mohawk
Contrapositive The contrapositive is formed by switching the order and making both negative. If tomorrow is not Thursday, If today is Wednesday, then tomorrow is Thursday. If today is not Wednesday, then 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 1.3conditional statements Continue lesson 1.3conditional 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 ifthen, 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