4.1 Proofs and Counterexamples

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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 . - PowerPoint PPT Presentation

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4.1 Proofs and Counterexamples

Even Odd NumbersFind a property that describes each of the following setsE={, -4, -2, 0, 2, 4, 6, }O={, -3, -1, 1, 3, 5, }

ExerciseUse the definitions of even and odd to

a. Justify why -4, 0, 8 are even numbers.b. Justify why 1, 11, 301 are odd numbersc. If a and b are integers, is 6a2b evend. If a and b are integers, is 10a + 8b + 1 odde. Which one of these statement is trueEvery integer is even and oddEvery integer is either even or oddPrime NumbersIf possible write the following positive numbers as a product of positive numbers less than the given number421818

Justifying Quantified StatementDetermine 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 numbersDisproof by Counterexample

Counterexample True False Proving StatementsThe 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 falseThe sum of two odd numbers is evenProvide several examples before you make a conjecture whether it is true or false

Indicate how you justify whether the statement is true or false

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 negationDIRECT PROOFSTo 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

Basic Results to masterMake a conjecture about each of the following results and then prove them directlyThe 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 _______.

The sum of any odd integer and any even integer is _______.

The difference of any odd integer minus any even integer is _______.

The difference of any even integer minus any odd integer is _______.