Proof notation Flashcards
How do you represent ODD NUMBERS in proof?
2n-1
How do you represent EVEN NUMBERS in proof?
2n
How do you represent 3 consecutive numbers in proof?
n-1, n, n+1
How to represent multiples of n in proof?
kn (5n)
What does squaring a number ensure in proof?
that its larger than 0
How to represent any two integers in proof?
n, m
How to represent a rational number in proof?
- a/b in it’s simplest form (a and b have no common factors)
- both a and b are integers
- where b != 0
How to represent a two digit number ab in proof?
10a+b where 1<=a<=9 and 0<=a<=9
What is a natural number’s symbol?
N
What is a natural number?
A number thats whole and >0
N={1,2,3,4…}
What is the symbol for an integer?
Z
What is the symbol for a positive integer?
Z+
What is an integer?
Any whole number
Z={0,+-1, +-2, +-3….}
What is the symbol for rational numbers?
Q
What does Q not include?
irrational numbers ( pi, surds etc.)
What is the symbol for real numbers?
R
What is a real number?
Any number, negative or positive, including irrational numbers
Steps for Proof by Contradiction
- Write the negation statement
- Use logical steps to show that the negation statement leads to a contradiction
- Write a conclusion- include the original statement
What is a negation statement?
A statement that asserts the falsehood of another statement.
What are the steps to prove by contradiction that √3 is an irrational number?
Suppose √3 is a rational number in the form a/b where a and b have no common factors
√3=a/b
3= a²/b²
3b²=a²
therefore a² is a multiple of 3
therefore a is a multiple of 3
a= 3n
a²= 9n²
3b²= 9n²
b²= 3n²
therefore b² is a multiple of 3
therefore b is a multiple of 3
both a and b have a common factor of 3
A contradiction
Therefore proving that √3 is an irrational number
How to use proof by contradiction to prove that there are infinately many prime numbers?
Assuming there is a finite number of prime numbers: p1, p2, p3… pn
Let P= p1xp2xp3….xpn+1
P is either a prime number or not a prime number:
If P is a prime number then it is not on the list.
If P is not a prime number then it has a prime factor, which must be a factor of the added one. This is impossible as 1 has no prime factors. Therefore this is a contradiction, proving there are infinately many prime numbers.
