4.1 Proofs and Counterexamples
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Transcript of 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.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 true
Every integer is even and oddEvery integer is either even or odd
Prime Numbers
If possible write the following positive numbers as a product of positive numbers less than the
given number421818
Justifying Quantified Statement
Determine the truth value of each statement below (Start by writing each statement and its negation
using symbolic logic)1. There exists an even number that is the sum of
two prime numbers
2. Any even number is the sum of two prime numbers
Disproof by Counterexample
Counterexample True
False
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
The 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 negation
DIRECT PROOFS
To prove directly that statement of the form
is true follow these steps : 1. Pick any x (but particular) in the domain and
assume P(x) is true (general-particular)2. Through a series of known true fact conclude
Q(x) is true
Basic Results to master
Make a conjecture about each of the following results and then prove them directly
1. The sum, product, and difference of any two even integers are _______.
2. The sum and difference of any two odd integers are ________.
3. The product of any two odd integers is _____.
4. The product of any even integer and any odd integer is _______.
6. The sum of any odd integer and any even integer is _______.
7. The difference of any odd integer minus any even integer is _______.
8. The difference of any even integer minus any odd integer is _______.