Section 1.3. Let I be a set. When I is used in the following context,

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1 Section 1.3. Let I be a set. When I is used in the following context, {B i } i I, we call I the index set. The set {B i } i I is the family of sets of the form B i where i I. One could also use set builder notation to indicate this same set, {B i i I}. If I = [n] (remember [n] = {1, 2,..., n}), we could denote the family as: {B 1, B 2,..., B n } or {B i i [n]}. When n = 1, {B 1, B 2,..., B n } is just {B 1 }. When n = 2, {B 1, B 2,..., B n } is just {B 1, B 2 }. When n = 3, {B 1, B 2,..., B n } is just {B 1, B 2, B 3 }. Definition 1. A partition of a nonempty set S is a collection C = {B i } i I of subsets of S that satisfy the following conditions. For each i I, B i C is a mutually disjoint family i I B i = S More notation: Another way to express i I B i is C or {B i } i I. Definition 2. If P = {B j j J} is a partition of a set S and each block B j is further partitioned into Q j = {C k k I j }, then the collection Q = Q of the blocks of the partition P forms a partition of the set S and we say the partition Q of S is finer than the partition P or that the partition Q of S is a refinement of P. We could also say that P is coarser than Q. Example 1. A partition of the integers into blocks, B 0, B 1, B 2, B 3 where B i are the integers which have a remainder of i upon division by 4. B = {B 0, B 1, B 2, B 3 } is a partition of Z. Example 2. A partition of the integers into blocks, A 0, A 1 where A i are the integers which have a remainder of i upon division by 2. What is another name for A 0? A 1? A = {B 0, B 1 } is a partition of Z. Is A a refinement of B? Or visa versa? 1

2 2 1.4: Logic and truth tables. 1.5: Quantifiers. 1.6: Implications. Definition 3. A mathematical statement must have a truth value, true or false. Example 3. For all x > 2, x 2 4x + 4 > 0. True or false? We use the symbol instead of the phrase for all. Example 4. x { 1, 2}, x 2 4x + 4 = 0. True or false? Example 5. There exists an x { 1, 2}, x 2 4x + 4 = 0. True or false? We use the symbol instead of the phrase there exists. Example 6. x > 2 such that 2 x = x 2. Example 7. The statement: x = is neither true nor false. Example 8. The statement: x = 1+ is neither true nor false. Example 9. The statement: This statement is false. is neither true nor false. Truth table for Negation. Given a statement P, the negation of P is denoted P or P. It is defined by the following truth table: P P T F We say P is false whenever P is true and true whenever F T P is false. We can use the truth table method to prove that ( P ) = P. P P ( P ) T F After filling in column 3, we see that column 3 F T and column one have the same truth value, so P = ( P ).

3 Compound Statements. Uses,, or. Conjunction: P Q is read as, P AND Q. Disjunction: P Q is read as, P OR Q. We use the following truth table to define these terms. P Q P Q P Q T T T T T F F T F T F T F F F F Simple Statements. Does not use,, or. Theorem 1. (1.4.1: De Morgan s Laws) (P Q) = ( P ) ( Q) and (P Q) = ( P ) ( Q). Proof using truth tables: P P Q Q ( P ) ( Q) P Q (P Q) T F T F T F F T F T T F F T F T P P Q Q ( P ) ( Q) P Q (P Q) T F T F T F F T F T T F F T F T 3

4 4 Theorem 2. (1.4.3: Distributive Laws) (P Q) R = (P R) (Q R) and (P Q) R = (P R) (Q R). Proof using truth tables: P Q R P Q (P Q) R P R Q R (P R) (Q R) T T T F T T F T T F T F T F F T F T T F F T F T F F T F F F F T P Q R P Q (P Q) R P R Q R (P R) (Q R) T T T F T T F T T F T F T F F T F T T F F T F T F F T F F F F T Exclusive OR. P or Q is true but not both. Compare to (P ( Q)) (Q ( P )) in a truth table. P P Q Q P XOR Q P ( Q) Q ( P ) (P ( Q)) (Q ( P )) T F T F F T F F T T F T T F T F T F T F

5 5 Quantifiers are used to describe variables in statements. - The universal quantifier means for all. - The existential quantifier means there exists. The phrases, for all x in R if x is an arbitrary element of R for every x in R let x be in R can be translated symbolically into: x R. The phrases, there exists an x in R such that for some x in R there is an x in R such that can be translated symbolically into: x R. More than one quantifier. We may say ( x E)( k Z)x = 2k or to mean the same thing, we could use commas x E, k Z, x = 2k Once a variable is quantified it is fixed for the remainder of the statement. For example, ( x R)( y R)( z R)x 2 + y 2 = z 2. This is a true statement. Proof: Let z = x 2 + y 2. The order of the quantifiers counts. ( y C)( x C)x = y 2 ( x C)( y C)x = y 2 The first is false and the second is true.

6 6 The negation of quantifiers. ( x, Q) = x, Q ( x, Q) = x, Q Example 10. Q: Every good boy does fine. Symbolic translation: x G, x does fine. Negation: x G, x does not do fine. Rewrite the negation in words: There are some good boys that do not do fine. Q Q ( P ) Notice that Q and its negation satisfy: T F F T Example 11. P: There is a broken chair in this room. Symbolic translation: x C, x is broken. Where C is the set of chairs in this room. Negation: x C, x is not broken. Rewrite the negation in words: All chairs in this room are not broken. Or, we might say, No chair in this room is broken. P P ( P ) Notice that P and its negation satisfy: T F F T Negation of double quantifiers. [ n N, x A, nx < 1] n N, [ x A, nx < 1] n N, x A, [nx < 1] n N, x A, nx 1

7 7 Implication - Implication. P Q P Q T T T T F F F T T F F T The phrases, P implies Q If P then Q Q if P (Make note of this one) Q is implied by P Q is true whenever P is true P only if Q can be translated symbolically into: P Q. In the implication P Q, P is the antecedent or hypothesis and Q is the consequence or conclusion. The Converse of P Q. is Q P. The Contrapositive of P Q. is ( Q) ( P ). Example 12. Is this statement true or false: x 2 = 4 if x = 2? Is its converse true or false? Example 13. P: it is sunny Q: there is a ball game The implication. P implies Q: If it is sunny then there is a ball game. Its converse: Q implies P: If there is a ball game then it is sunny. In this case, we could also say, There is a ball game only if it is sunny. Its contrapositive: not Q implies not P: If there is no ball game then it is not sunny. There is a ball game only if it is sunny.

8 8 The implication, P Q, and its contrapositive, Q P, have the same truth value. Lets look at the truth table. P Q P Q ( Q) ( P ) P Q T T F F T T T F F T F F F T T F T T F F T T T T Example 14. Show (P Q) has the same truth value as P ( Q). Example 15. P: If x < 0 then x = x. Negate P. Example 16. Q: f(x) = f(y) implies x = y. Q is neither true nor false because it is not quantified. We can quantify it as follows: For x R, and f(x) = 3x 2, Q. or For x R, and f(x) = x 2, Q. Is the first example true or false? Is the second example true or false? Example 17. Negate both statements from the previous example. P Q is said to be vacuously true when P is false. For example, If a snowball survives in hell, I will eat my hat. Example 18. Use a truth table to prove that ((P Q) (Q R)) (P R) For the following two examples, let O be the odd integers and E be the even integers. Example 19. Determine if the following is true or false. xy O (x O y O) Example 20. Determine if the following is true or false. xy E (x E y E)

9 9 If and only if statements. P Q P Q T T T T F F F T F F F T The phrases, P if and only if Q If P then Q and if Q then P P is a necessary and sufficient condition for Q. can be translated symbolically into: P Q. P Q is the only if part, also called the sufficiency. That is P is a sufficient condition for Q. P Q is the if part, also called the necessity. That is P is a necessary condition for Q. Example 21. Let P and Q be statements. Prove that the following are true. (Q Q) P. Q P Q (Q Q) (Q Q) P T T F F T T F F F T F T T F T F F T F T P Q P. P Q P Q P Q P T T T T T F F T F T F T F F F T P P Q. P Q P Q P P Q T T T T T F T T F T T T F F F T

10 10 Some word problems: Example 22. The following sentence appeared on a restaurant menu: Please note that every alternative may not be available at this time. Describe the unintended meaning. Rewrite the sentence to state the intended meaning clearly. Unintended meanings: Rewrite: Please note that some alternatives may not be available at this time. Not every option is available currently Please note that not every alternative may be available at this time. Every offered alternative will be available later. Please note that alternative items are subject to availability. Example 23. Give an example of an English sentence that has different meanings depending on inflection, pronunciation, or context. Everyone is wrong some of the time I lie here often. Example 24. From outside mathematics, give an example of statements A, B, C such that A and B together imply C, but such that neither A nor B alone implies C. A: I love you. B: You love me. C: We love each other. A: Dog is hungry. B: Food is put out. C: Dog is eating. A: Person x is a natural born US Citizen. B: Person x is 38 years old C: Person x can be president of the US A: It is below 32 degrees. B: there is precipitation. C: It will snow. A: She is a good teacher. B: She is a good mathematician. C: She is a good math teacher. Example 25. Negate the statement no slow learners attend this school. Sol: P: x S, x A. P : x S, x A. In words, the negation is, there is a slow learner in this school.

11 Example 26. A fraternity has a rule for new members: each must always tell the truth or always lie. They know who does which. If I meet three of them on the street and they make the statements below, which ones (if any) should I believe? A) Says: All three of us are liars. B) Says: Exactly two of us are liars C) Says: The other two are liars. 11

12 12 Examples from 1.4, 1.5, and Express each of the following statements as a conditional statement in if-then form or as a universally quantified statement. Also write the negation. (1) Every odd number is prime. ( x odd integers ) x is prime. Negation: ( x odd integers ) x is not prime. (2) The sum of the angles of a triangle is 180 degrees. If x, y, z are the angles of a triangle, then their sum is 180 degrees. Negation: x, y, z are the angles of a triangle and their sum is not 180 degrees. (3) Passing the test requires solving all the problems. If you passed the test then you solved all the problems. Negation: You passed the test and you did not solve all the problems. (4) Being first in line guarantees getting a good seat. If you are first in line, then you will get a good seat. Negation: You are first in line and you do not get a good seat. (5) Lockers must be turned in by the last day of class. ( x lockers) x is turned in by last day of class. Negation: ( x lockers ) x is not turned in by the last day of class. (6) Haste makes waste. ( x haste) x makes waste Negation: ( x haste ) x does not make waste. (7) I get mad whenever you do that. If you do that then I get mad. Sometimes you do that and I do not get mad. (8) I won t say that unless I mean it. If I mean that then I will say that. Sometimes I won t say that and I mean that. 2. Which of these statements are believable? (1) All of my 5-legged dogs can fly. true (2) I have no 5-legged dog that cannot fly. true (3) Some of my 5-legged dogs cannot fly. false (4) I have a 5-legged dog that cannot fly. false 3. Prove that if x and y are distinct real numbers, then (x + 1) 2 = (y + 1) 2 if and only if x + y = 2.

13 Proof: We start with (x + 1) 2 = (y + 1) 2. Expanding both sides, we get x 2 + 2x + 1 = y 2 + 2y + 1. Canceling the 1 s and rearranging terms, we get x 2 y 2 = 2y 2x. We factor each side to get (x + y)(x y) = 2(y x) = 2(x y). As we are given x y, we can divide both sides by x y to get x + y = 2. As equations preserve equality and these steps are reversible, x + y = 2 implies (x + 1) 2 = (y + 1) 2, so the dependence holds both ways. 4. Given a real number x, let A be the statement 1/2 < x < 5/2, let B be the statement x Z, let C be the statement x 2 = 1, and let D be the statement x = 2. Which statements below are true for all x R. (1) A C. False. For a counterexample, let x = 2. (2) B C. False. For a counterexample, let x = 2. (3) (A B) C. False. For a counterexample, let x = 2. (4) (A B) (C D). True. If A and B are true, then x = 1 or x = 2. If x = 1 then C is true. If x = 2 then D is true. So in either case, either C or D is true. (5) C (A B). False. For a counterexample let x = 1. (6) D [A B ( C)]. True. If x = 2 then A is true, B is true, and x 2 1 so C is true. (7) (A C) B. True. A C means that x = 1, 2, or 1. So, B is true. 5. Let x, y be integers. Determine the truth value of each statement below. (1) xy is odd if and only if x and y are odd. TRUE. If x is odd and y is odd then x = 2a + 1 for some a and y = 2b + 1 for some b. Then xy = (2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1 which is odd. Suppose it is not true that both x and y are odd. Then at least one is even. Without loss of generality, suppose x is even. Then x = 2a for some a. xy = 2ay which is even. (2) xy is even if and only if x and y are even. FALSE. For a counterexample to the only if part, let x = 3 and y = 4. Then, xy = 12 which is even. 13