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Mathematical Thinking 1215 Week of 1/9/12
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1 The Quick Reference Guide •
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Logic (pg.5) A branch of mathematics that concerns itself with basic assumptions of mathematics and the structure of proofs. Statement (pg.5)—a declarative sentence that has either a true or false truth-­‐value. Compound Statement (pg.6): A statement formed by combining two or more statements using various logical connectives such as and, or, if, and then. ^ = and, v = or For a compound statement with ‘and’ to be true, both components have to be true. Only one has to be true for a compound statement with ‘or.’ A Truth Table (pg.7) is a table of all the possibilities for the truth-­‐
value of a statement whose truth-­‐value is unknown. Negation (pg.8): Negation of a statement is the opposite of that statement. Not = ~ (negation) A truth table with two variables has four rows. A truth table with three variables has eight. Equivalent Statements: Two compound sentences are logically equivalent if they have the same truth-­‐value for all possible assignments of truth-­‐value. DeMorgan’s law (pg.13) states that when a negation is distributed within a parentheses, ^ becomes v and vice versa. The Conditional Statement (pg.13) is a compound statement that uses the connective then. “If …p…then …q” pq Laws of compound statements (pg.14): Commutative law Associative law Distributive law DeMorgan’s law Absorption law notely
2 Mathematical Thinking First Class h.sadaka@neu.edu Syllabus handed out Link to Syllabus: http://blackboard.neu.edu/bbcswebdav/pid-­‐3855875-­‐dt-­‐content-­‐rid-­‐
4541567_1/courses/MATH1215.30197.201230/syllabus_1215Spring2012_Section
1%281%29.pdf You will need your myNEU login info to view this syllabus. Class registration logistics: The course book is only available at northeastern. The book is optional, because the online access code includes the electronic book. To buy the access code online, go to http://www.coursecompass.com and register as a student. Enter the course ID: sadaka48589. If you’ve already used coursecampus, enter your username and password. Otherwise, create an account. Pay with a credit or debit card. The homework will not be on Blackboard, it will be at http://www.coursecompass.com, which is where the MyMathLab tools for homework will be located. Before beginning work, you need to set up your student account. All the homework is due the last day of class, so it is your responsibility to go over it once the instructor has covered the material for the assignments in class and to stay up to date. Press the ‘Clear All’ button on the lower right hand corner of the homework questions to redo your answer. notely
3 There will be at least one in-­‐class problem per week. You may discuss these with a partner, and sometimes they will be collected. These problems are like attendance—if you get the answer wrong, you will still get the 5% credit for the in-­‐
class problems. Quizzes will be given in class, and the best seven quizzes will be counted. There will be 9~12 quizzes. Logic Chapter 2-­‐3: Probability—sets, counting technique, basic rules, probability distribution (discreet). Statistics—normal distribution Definitions: Logic. A branch of mathematics that concerns itself with basic assumptions of mathematics and the structure of proofs. Sentences. There are many kinds of sentences— •
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Interrogative: ‘How old are you?’ Imperative: ‘Shut the door.’ Declarative: ‘Today is Monday.’ Subjunctive: ‘I wish I were rich.’ Section 6.1 Statement—a declarative sentence that has either a true or false truth-­‐value. We use shorthand to write this. notely
4 V(p) = T or F V stands for the truth-­‐value, and p represents the statement. q, r, and s are also used when more than one statement is being dealt with. Logic concerns itself with statements. Ex: Tomorrow is Wednesday. … False The sky is green. … False 1+1 = 3 … False 11+ 5 = 16 … True It rained yesterday in Miami … unknown If the truth-­‐value of the statement is unknown, then there are two possibilities for its truth value: True or False. Compound Statement: A statement formed by combining two or more statements using various logical connectives such as and, or, if, and then. We use symbols for these logical connectives: ^ = and v = or Ex: Descrates was a philosopher and a mathematician. Statement p = Descrates was a philosopher. (This is true) Statement q = Descrates was a mathematician (This is true) Therefore, V(p ^ q) = T notely
5 When a compound statement is created using and, both statements within the compound statement must be true in order for the compound statement to be true. When a compound statement is created using or, only one of the component statements must be true in order for the compound statement to be true. Take for example p^q. If we don’t know the truth-­‐values of the component statements, then there are four possible overall outcomes. This is because there are two possible outcomes for each individual statement, and there are two individual statements. Two multiplied by two equals four. p q p^q T T T T F F F T F F F F This table is called a ‘Truth Table.’ It is a table of all the possibilities for the truth-­‐
value of a statement whose truth-­‐value is unknown. Now let’s look at the statement p v q: p q p v q T T T T F T F T T F F F Only the compound statement made up entirely of false statements is considered false. notely
6 Some more examples of compound statements: My child is a boy or a girl. (The statements ‘My child is a boy.’ and ‘My child is a girl. ‘ are combined with ‘or’ to create a compound statement.) = p v q The weather is cold and rainy. = p^q He likes running or biking. = p v q We return to logical statements, compound sentences, and truth tables. Compound statements: We begin with the statement: Today is Wednesday. This is true, therefore, V(P) = T We can combine ‘Today is Wednesday’ with ‘It is cold today’ with the logical connective ‘and,’ because both statements are true. ‘Today is Wednesday and it is cold.’ This is a compound statement. V(P^Q) = T Symbols and statement types for logical connectives: And = ^ conjunction Or = v disjunction Negation: Negation of a statement is the opposite of that statement. If a statement is negative, it is given a negation sign. Not = ~ (negation ) The negation sign comes before the statement itself … ~(P^Q) or ~P. For the statement ‘Today is not Wednesday,’ we write V(~P) = F, because we know that today IS Wednesday. The truth-­‐value for the statement ~P is therefore false. notely
7 Review of compound statements using and and or: Statement (P^Q) is a statement that is always true when both P and Q are true, and false otherwise. Statement (PvQ) is a statement that is always true when at least P or Q is true, and false otherwise. Truth Table for (P^Q): P Q P^Q T T T T F F F T F F F F Notice that the first column alternates T and F by two, and the second alternates T and F by one. Truth Table for (PvQ): P Q PvQ T T T T F T F T T F F F Truth tables for more complex compound statements: Truth Table for P^(QvR): There will be eight outcomes, since P, Q, and R have two outcomes each. 2x2x2 = 8 P Q R QvR P^(QvR) T T T T T T T F T T T F T T T notely
T F F F F F T T T F F T F T F F F T T F F F F F F Notice that the first column alternates T and F by four, the second alternates T and F by two, and the third alternates T and F by one. Truth tables using negation: Truth Table for pv~p: There will be two outcomes, because there is only one letter. p ~p p v~p T F T F T T Therefore, V(pv~p) = T. We call this tautology, when all the truth-­‐values of a combined statement are true no matter what truth-­‐values are assigned to the components. Consequentially, the truth-­‐value of statement p^~p is always false. This is called a contradiction—the outcome is always false no matter what truth-­‐values are assigned to the components. p ~p p^~p T F F F T F Determining the truth-­‐values of compound statements using given truth-­‐
values for individual statements: If P is false, Q is true, and R is false: 8 notely
9 ~Qv(~P^R) ~Tv(~F^F) Fv(T^F) FvF F (~RvQ)^(~QvP) (~FvT)^(~TvF) (TvT)^(FvF) T^F F Negation in written statements: Let t= ‘It is snowing today’ u= ‘I won the lottery on Monday’ t^u = ‘It is snowing today and I won the lottery on Monday’ tv~u= ‘It is snowing today or I did not win the lottery on Monday’ ~(tvu)= ‘It is not the case that it is snowing today or that I won the lottery on Monday.’ Complex truth tables: We write the truth-­‐values first for the individual statements, then for the statements within parentheses, and finally for the entire compound statement. Truth table for ~(~t) t ~t ~(~t) T F T notely
F T F Truth table for (Pv~Q)^Q P Q ~Q (Pv~Q) (Pv~Q)^Q T T F T T T F T T F F T F F F F F T T F Truth table for (~PvR)^(QvP) P Q R ~P (~PvR) (QvP) (~PvR)^(QvP) T T T F T T T T T F F F T F T F T F T T T T F F F F T F F T T T T T T F T F T T T T F F T T T F F F F F T T F F 10 notely
11 Equivalent statements and DeMorgan’s law: Two compound sentences are logically equivalent if they have the same truth-­‐value for all possible assignments of truth-­‐value. Are the statements ~(p^q) and ~pv~q equivalent? Yes. We use DeMorgan’s law, which states that when a negation is distributed within a parentheses, ^ becomes v and vice versa. For example: Negation of p^~q is ~(p^~q) , which equals ~pvq when DeMorgan’s law is used. Negation of p^(~qvp) is ~(p^(~qvp) , which equals ~pv(q^~p) when DeMorgan’s law is used. Using DeMorgan’s law to negate written compound statements: I earned an A or I earned a B. I did not earn an A and I did not earn a B. I love cars and I love money. I do not love cars or I do not love money. Conditional statements: The conditional statement is a compound statement that uses the connective then: “If …p…then …q” Statement p is called the antecedent, and statement q is called the consequent. Example: A political candidate says “If I am elected, then no new taxes will be introduced.” Let p = he is elected. notely
q = there are no new taxes. Truth table: Elected no new taxes statement P q pq T T T T F F F T T F F T If (p~q) (~rq) and p, q, and r are all false, what is the truth-­‐value? (F T) (T F) T F F Laws of compound statements: Commutative law: When the signs are the same, the variables can be re-­‐ordered. p^q = q^p Associative law: When the signs are the same, the brackets can be shifted. pv(qvr) = (pvq) vr Distributive law: 12 notely
When the signs are opposite, the brackets can’t be shifted but the statement outside the parentheses can be distributed. pv(q^r) = (pvq )^ (pvr) DeMorgan’s law: The negation is distributed and the sign is switched. ~(p^q) = ~pv~q Absorption law: When the connectives are different, the ‘p’s absorb the ‘q.’ (p^q) v p = p We can make a truth table to prove any of these laws. With regards to the law of absorption, we see on a truth table that the p column and the (p^q) v p column are exactly the same: p q p^q (p^q) v p T T T T T F F T F T F F F F F F Negation of a conditional as an ‘or’: We take for example the statement p q: ~(p q) = ~p v q the negation is distributed, and the then becomes or. Contrapositive statements: Take for example the statement ‘If it is raining, then the grass will be green.’ The contrapositive statement would then be ‘If the grass is not green, then it is not raining.’ 13 notely
In our example, both the original and the contrapositive statements are true. The contrapositive of a statement will always have the same truth-­‐value as the original statement. For the statement p q , the contrapositive statement is written as ~q ~p . Notice the switch of the letters: looking at the written statement example, we see that the state of the weather and the color of the grass are also switched in the contrapositive. 14