5. Use a truth table to determine whether the two statements are equivalent. Let t be a tautology and c be a contradiction.

Transcription

1 Statements Compounds and Truth Tables. Statements, Negations, Compounds, Conjunctions, Disjunctions, Truth Tables, Logical Equivalence, De Morgan s Law, Tautology, Contradictions, Proofs with Logical Equivalent Table and Truth Tables, Conditionals, Converses, Inverses, and Contrapositives. Be able to: Identify Statements Negate statements, conjunctions, disjunctions, conditionals, ect. Construct truth tables Write a proof using the Logical Equivalent rules Write and identify conditionals and all forms of conditionals. (converse inverse and contrapositive) Problems: 1. Indicate which sentences are statements. a. 2 is the smallest prime number. b. Math is awesome. 2. Fill in the blanks so that argument (b) has the same form as argument (a). Represent each form using letters as a guide. a. If all integers are rational, then the number 1 is rational. All integers are rational. Therefore, the number 1 is rational. b. If all integers are closed under addition, then?.?. Therefore, is an integer. 3. Write the statements in symbolic form using the symbols ~ and the indicated letters to represent component statements. Let s = stocks are increasing and i = interest rates are steady a. Stocks are increasing but interest rates are steady. b. Neither are stocks increasing nor are interest rates steady. 4. Write the truth tables for the statement: p q ~ p q 5. Use a truth table to determine whether the two statements are equivalent. Let t be a tautology and c be a contradiction. p q r p q p r a. and b. p c and p 6. Use De Morgan s Law to write the negations for each statement. Assume x is a particular real number.

2 a. Integers are rational and imaginary numbers are not real. b. 4 x 2 7. Determine which statement forms are tautologies and which are contradictions. p ~ q ~ p q a. b. ~ p q p ~ q 8. Prove the following are logically equivalent by using the Logical Equivalence Table. a. p ~ q p p 9. Rewrite the statements in If then form a. Study or you will fail. b. A necessary condition for the sum of x and y to be an integer is that x and y both be integers. 10. Construct the truth table for each statement: a. p ~ p q q 11. Use a truth table to verify the following statement is true: a. p q ~ p q 12. Write the negation, converse, inverse and contrapositive for the statement: a. If m and n are integers, then the sum of m and n is an integer.

3 Topics: Biconditionals, Arguments, Digital Logic Circuits and Number Systems. Be able to: Identify a biconditional, construct the truth table. Show and prove arguments are valid/invalid using truth tables or valid argument forms (modus ponens, tollens ect) Apply the contradiction rule. Construct the input/output table of a given circuit. Give the Boolean expression of a circuit, construct a circuit with a given Boolean expression, use Scheffer strokes and Pierce arrows. Convert decimal integers to binary and vice versa, add and subtract binary digits. 13. Construct a truth table for the following biconditional: p ~ r q r a. 14. Use modus ponens or modus tollens to fill in the blanks in the arguments so as to produce a valid argument. a. If a divides b then a and b are integers. a divides b. Therefore. b. If the product of two number is rational, then the two numbers are real. The two numbers are not real. Therefore. 15. Determine whether the arguments are valid or invalid by using a truth table. p q p q a. q p p q b. q p 16. Write each of the following symbolically and determine if the argument is valid or invalid. Identify the rule that makes the argument valid or invalid. a. x is rational or it is irrational. x is not rational. Therefore x is irrational. 17. Give an example of an invalid argument that has a true conclusion. 18. Give an example of a valid argument with a false conclusion. 19. Knights always tell the truth and Knaves always lie. Person A says: Both of us are knights. Person B says: A is a knave. What kind of people are A and B? 20. Write an input/output table for the given circuit. Give the Boolean expression for the circuit.

4 21. Construct circuits for the following Boolean Expressions: P ~ P Q a. b. P Q ~ R 22. Give an equivalent circuit with at most two logic gates for the following: P Q ~ P Q ~ P ~ Q a. 23.Show that for the Scheffer stroke, p q p q p q 24. Show that the following are logically equivalent for the Peirce arrow, where p q ~ p q a. ~ p p q 25. Represent the following in binary notation. a. 287 b Represent the following integers in decimal notation a b Perform the arithmetic in binary notation: a b

5 Topic: Quantified Statements Quantified Statements and their truth sets. Implicit Quantification Quantified statements: Universal Statements, truth values and negations Existential Statements, truth values and negations Formal and Informal forms of quantified statements. Equivalent forms of quantified statements. ie All A are B There are No A that are not B ect. Multiple Quantifiers and their negations. Universal Conditional Statements, Converse, inverse and contrapositive. Arguments with quantified statements: Valid forms - Universal Modus Ponens and Tollens Invalid forms: Converse and Inverse errors. 28. Let Q ( n) n 2 50 a. Find Q(n) for n = {2, -10, 13, -20} and indicate which is true or false. b. Find the truth set of Q(n) if the domain is Z. 29. Let R be the domain of x. Let P ( x) x 2, Q ( x) x 4, R ( x) x 2, T ( x) x 2 2 a. Give the truth sets for each predicate. b. Give all the implicit quantification,, for each pair of predicates 30. Let Q( x, y) y 2 2 If x y then x with a domain for both x and y being R. a. Give an example that makes Q(x,y) false. b. Give example which makes Q(x,y) true. 31. Find the truth set for the given predicate a. P ( x) 1 x 2 4 where the domain of x is R. 32. Give a counterexample to show the statements are false: a. x, y R, x y x y 33. i. Express the quantified statement in an equivalent way. ii. Write the negation of the given quantified statement. a. All whales are mammals. b. Some journalists are writers. c. No circles are polygons. d. Some triangles are not equilateral. e. No perfect squares are imaginary.

6 34. Negate the following statements a. The sum of two integers is an integer. b. For all integers n, if 2 n is even then n is even. 35. Let D = {-48, -14, -8, 0, 1, 3, 16, 23, 26, 32, 36} Determine which of the following statements are true and which are false. Provide counterexamples for those statements that are false. a. x D, if x is odd, then x > 0 b. x D, if x is less than 23 then x is even. 36. Write the negation of the statement. a. x R, if x > 5 then x real numbers x, an integer n such that n > x a. Prove that this statement is true. (give an explanation) b. For each x given below, find an n to make the predicate n > x true. i. x = ii. x = 10² 38. Rewrite the statement without using or and b.) write the negation of the statement. a. odd integers n, an integer k such that n = 2k + 1. b. r Q, integers a and b such that r = a/b. 39. Rewrite the statement formally using quantifiers and variables. b) write a negation for the statement. a. All integers are rational. b. The square of all nonzero integers is positive. 40. Use universal instantiation or universal modus ponens to fill in valid conclusions for the following: If an integer n equal 2 k + 1 and k is an integer, then n is odd. 9 equals 2 4 1and 4 is an integer Use universal instantiation or universal modus tollens to fill in valid conclusions for the following: All healthy people eat an apple a day. Harry does not eat an apple a day..

7 42. Write each of the following formally with symbols,. Determine whether each argument is valid by universal modus ponens or universal modus tollens or invalid by the converse or inverse error. a. All healthy people eat an apple a day. Helen eats an apple a day. Helen is a healthy person. b. All honest people pay their taxes. Dan is not honest. Dan does not pay his taxes. 43. Indicate with of the following are valid or invalid. Support your answer with a diagram. a. All dolphins are mammals. All mammals are human. All dolphins are human.

8 Topic: Direct Proofs and Counter Examples: Integers, Rationals, Divisibility and Quotient Remainder Theorem. Definitions: Even, Odd, Prime, Composite, Proving Existential and Universal Statements, Method of Exhaustion, Method of Direct Proof, Common Mistakes in writing proofs, Disproof by Counterexample, Definitions: Rational, irrational, divisibility Theorems: Integers are rationals, the sum or rationals is rational, transitivity of divisibility, divisibility by a prime, unique factorization theorem (in standard form), Sieve of Eratosthenes, Divisibility by 2-12 x Definitions: Quotient remainder theorem, proof by cases, n div d, n mod d, x and 44. Assume that m and n are particular integers. Justify your answers to each of the following questions. a. Show 8m + 6n is even b. Show 12mn + 5 is odd. 45. Prove the statements a. There is an integer n > 5 such that 2n + 4 is divisible by 10. b. There are integers a and b and c such that 46. Prove the following by method of exhaustion. Every positive integer between 2 and 25 can be expressed as a product of primes. a 2 b 2 c 47. Prove the following or give a counterexample. a. The sum of an odd and an even is odd. b. If n is any odd integer, then (-1) n = -1 c. The product of any even integer and any integer is even. d. The difference of any two odd integers is odd. 48. Prove the following are rational numbers. a b Prove or disprove (with a counterexample) each of the following problems. Write all proofs in the correct format. a. The difference of any two rational numbers is a rational b. The product of any two rational numbers is rational c. The square of any rational is rational. d. The sum of a rational and an integer is an integer. 2

9 50. Give a reason for the following questions. Assume all variables are integers. a. Is (3k +1) (3k + 2)(3k + 3) divisible by 3? b. Does ? c. Is 6a(a + b) a multiple of 2a? 51. Prove the following statements if they are true or give a counterexample if it is false. Write all proofs in the correct format. a. The product of any two even integers is a multiple of 4. b. For all integers a, b and c, if a is a factor of c then ab is a factor of c. c. For all integers a, b, and c, if a (b + c) then a b and a c. d. The sum of three consecutive integers is divisible by Check the following integers for divisibility by 2-12 a b Write the following integers in standard factored form. a. 12! b For each of the values of n and d given, find integers q and r such that n = dq + r and 0 r d a. n = 82, d = 4 b. n = 7, d = Evaluate the expressions for the following. a. 47 div 6 b. 47 mod 6 a. 29 div 4 b. 29 mod Prove the following by breaking the problem into cases. a. The product of any two consecutive integers is even. b. Show that any integer n can be written in one of the four forms: n = 4q, n = 4q + 1, n = 4q + 2, or n = 4q + 3 for some integer q. 57. Compute x and x for each values of x. a b c. -24/7 d. 12/5 58. Use the floor notation to express each of the following: a. 25 div 7 and 25 mod 7 x y x y 59. Prove for all real numbers x and y, 60. For all real numbers x, x 2 x 2