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4.1 Proofs and Counterexamples

4.1 Proofs and Counterexamples. Even Odd Numbers. Find a property that describes each of the following sets E={ …, -4, -2, 0, 2, 4, 6, …} O={ …, -3, -1, 1, 3, 5, …}. Exercise. Use the definitions of even and odd to a . Justify why -4, 0, 8 are even numbers .

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4.1 Proofs and Counterexamples

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  1. 4.1 Proofs and Counterexamples

  2. Even Odd Numbers Find a property that describes each of the following sets • E={…, -4, -2, 0, 2, 4, 6, …} • O={…, -3, -1, 1, 3, 5, …}

  3. Exercise Use the definitions of even and odd to a. Justify why -4, 0, 8 are even numbers. b. Justify why 1, 11, −301 are odd numbers c. If a and b are integers, is 6a2beven d. If a and b are integers, is 10a + 8b + 1 odd e. Which one of these statement is true Every integer is even and odd Every integer is either even or odd

  4. Prime Numbers If possible write the following positive numbers as a product of positive numbers less than the given number 4 2 1 8 18

  5. Justifying Quantified Statement Determine the truth value of each statement below (Start by writing each statement and its negation using symbolic logic) • There exists an even number that is the sum of two prime numbers • Any even number is the sum of two prime numbers

  6. Disproof by Counterexample Counterexample True False

  7. Proving Statements The sum of two even numbers is even Provide several examples before you make a conjecture whether it is true or false Indicate how you justify whether the statement is true or false

  8. The sum of two odd numbers is even Provide several examples before you make a conjecture whether it is true or false Indicate how you justify whether the statement is true or false

  9. Discuss the truth value of There is a positive integer n such that n2 + 3n + 2 is prime HINT: a. Analyze different cases b. Look at its negation

  10. DIRECT PROOFS To prove directly that statement of the form is true follow these steps : • Pick any x (but particular) in the domain and assume P(x) is true (general-particular) • Through a series of known true fact conclude Q(x) is true

  11. Basic Results to master Make a conjecture about each of the following results and then prove them directly • The sum, product, and difference of any two even integers are _______. • The sum and difference of any two odd integers are ________. • The product of any two odd integers is _____. • The product of any even integer and any odd integer is _______.

  12. The sum of any odd integer and any even integer is _______. • The difference of any odd integer minus any eveninteger is _______. • The difference of any even integer minus any odd integer is _______.

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