Chapter 1 Flashcards
What is proof?
Proof is a series of logical steps which show whether a result is true or not for a set of specified numbers
e.g. All integers, all even numbers etc
A proof is a logical and structured argument to show that a mathematical statement (or conjecture) is always true.
A mathematical proof usually starts with previously established mathematical facts (or theorems) and then works through a series of logical steps. The final step in a proof is a statement of what has been proven.
An example of deduction
Here is an example of proof by deduction:
You can prove a mathematical statement is true by deduction. This means starting from known facts or definitions, then using logical steps to reach the desired conclusion.
Statement:
The product of two odd numbers is odd.
Demonstration:
5×7=35, which is odd.
(This is a demonstration, but it is not a proof. You have only shown one case.)
Proof:
Let p and q be integers, so 2p+1 and 2q+1 are odd numbers.
(You can use
2p+1 and 2q+1 to represent any odd numbers. If you can show that
(2p+1)×(2q+1) is always an odd number, then you have proved the statement for all cases.)
(2p+1)×(2q+1)=4pq+2p+2q+1 = 2(2pq+p+q)+1
Since p and q are integers, 2pq+p+q is also an integer.
Since 2(2pq+p+q)+1 is one more than an even number, it must be odd.
So the product of two odd numbers is an odd number.
This is the statement of proof.
What does a mathematical proof usually start with?
A mathematical proof usually starts with previously established mathematical facts (or theorems) and then works through a series of logical steps. The final step in a proof is a statement of what has been proven.
What does the N sign mean?
Natural numbers are only positive whole numbers, starting from 1, 2, 3, 4, … and so on. They do not include negative numbers or zero.
the letter ℤ what does it stand for in maths?
the letter ℤ represents the set of integers.
The Z comes from the German word “Zahlen,” which means “numbers.”
So, ℤ includes all positive and negative whole numbers, including zero.
What does the letter ℚ mean?
In mathematics, the letter ℚ represents the set of rational numbers.
ℚ (Q) represents the set of rational numbers:
ℚ = { a/b | a, b ∈ ℤ, b ≠ 0 }
(Any number that can be written as a fraction)
(Comes from the word “Quotient”)
In mathematics, What does ℝ (R) represent?
In mathematics, ℝ (R) represents the set of real numbers.
Definition:
ℝ = { all rational and irrational numbers }
This includes:
Rational numbers (ℚ): Numbers that can be written as fractions (e.g., 1/2, -3, 0.75, 7).
Irrational numbers: Numbers that cannot be written as fractions (e.g., π, √2, e).
Example of real numbers:
-2, 0, 3.5, π, √2, 10
Real numbers do not include imaginary numbers (like √-1, also called i).
What are some key mathematical concepts linked to proof techniques?
- Odd numbers: 2n - 1
- Even numbers: 2n
- Three consecutive integers: n-1, n, n+1
- Squares: Anything squared is always greater than or equal to zero
- Multiples of integers: kn for some integer k
True or False
Integers are used frequently in the language of proof
The set of integers is denoted by Z.
True
Q: What does P ⇔ Q mean?
A: P is equivalent to Q, true if and only if both are true.
What is proof by deduction?
Proof by deduction is when a mathematical and logical argument is used to show whether or not a result is true
How to Prove Something Using Deduction
How to Prove Something Using Deduction
If a question asks you to prove something using deduction, follow these steps:
- State the Given Information – Identify the known facts.
- Use Definitions and Theorems – Apply mathematical rules logically.
- Show Step-by-Step Reasoning – Each step must follow logically from the previous one.
- Arrive at the Conclusion – Prove the required result logically.
Example: Prove that the sum of two even numbers is always even using deduction.
Step 1: Define even numbers
- Let the two even numbers be 2a and 2b, where a, b are integers.
Step 2: Add the numbers
- ( 2a + 2b = 2(a + b) ).
Step 3: Show the result is even
- ( 2(a + b) ) is a multiple of 2, so it is even.
Step 4: Conclusion
- Since the sum of two even numbers is always a multiple of 2, it is always even. ✅
This is deductive reasoning because we used definitions and logical rules to reach the conclusion.
How to prove a proof using exhaustion?
Break down the statement into smaller cases and prove each case separately.
This method is better suited to a small number of results. You need many examples to prove a statement is true, as one example is only one case.
What you must include in a mathetmatical proof?
- State any information or assumptions you are using
- Show every step of your proof clearly
- Make sure that every step follows logically from the previous step
- Make sure you have covered all possible cases
- Write a statement of proof at the end of your working
Theorem
A statement that has been proven
Conjecture
a statement that has yet to be proven
How to prove results involving identites, such as (a + b)(a - b) ≡ a² - b²
To prove an identiy you should
- Start with expression on one side of the identity
- Manipulate that expression algebraically until it matches the other side
- Show every step of your algebraic working
Q: How are even and odd numbers represented in proofs?
A: Even numbers are represented as 2n and odd numbers as 2n - 1.
Q: What sets of numbers should you be familiar with?
A: N (natural numbers), Z (integers), Q (rationals), and R (real numbers).
Q: How do you show an expression is a multiple of a number?
A: Write multiples of n in the form kn, where k is an integer.
Q: What should be used in proofs for algebraic simplification?
A: Use algebraic techniques and show logical steps with correct mathematical notation.
Define the term rational number
any number that can be expressed as the ratio of two integers, where the denominator is not zero (e.g., 1/2, -3/4, 5).
is any number that can be written as a fraction, like 1/2 or 3/4.
Define the term irrational number
Irrational number
a number that cannot be expressed as a simple fraction, and its decimal form never repeats or ends (e.g., √2, π, e).
What are the relationship between the sets? N ⊆ Z ⊆ Q ⊆ R.
R (Real numbers) contains all numbers on the number line, including fractions, decimals, and irrationals.
Q (Rational numbers) is a subset of R, containing numbers that can be expressed as fractions.
Z (Integers) is a subset of Q, containing whole numbers and their negatives, as well as zero.
N (Natural numbers) is a subset of Z, containing counting numbers starting from 1.
Prove that the square of an even number is also even:
Let the even number be 2n.
(2n)² ≡ 4n²
≡ 2(2n²)
“Even” is the same as “multiple of 2.”
≡ 2k for some integer k.
So, the square of an even number is even.
What is proof by exhaustion?
Proof by exhaustion is a way to show that the desired result works for every allowed value
How do I prove a result by exhaustion?
Using proof by exhaustion means testing every allowed value not just showing a few examples
what are the difficulties with proof by exhaustion?
Complexity: Exhaustive cases can become very time-consuming and complex, especially when the number of cases increases.
Oversight: It’s easy to overlook a case or make errors in individual case analysis.
Limited Applicability: This method is not suitable for infinite or very large sets of cases, making it impractical in some situations.
Prove that the difference between n³ + 7 and a multiple of 7 is always 1, for all integers in the interval 1 ≤ n ≤ 4.
Prove that the difference between n³ + 7 and a multiple of 7 is always 1, for all integers in the interval 1 ≤ n ≤ 4.
Note: This result may be true for all values of n but you are only asked to prove it for integers in the interval 1 ≤ n ≤ 4, i.e. n=1, n=2, n=3, n=4.
So ‘Exhaust’ all possibilities.
For n = 1:
1³ + 7 = 1 + 7 = 8
Difference of 1 from 7.
For n = 2:
2³ + 7 = 8 + 7 = 15
Difference of 1 from 14.
For n = 3:
3³ + 7 = 27 + 7 = 34
Difference of 1 from 35.
For n = 4:
4³ + 7 = 64 + 7 = 71
Difference of 1 from 70.
Conclusion: By exhaustion, n³ + 7 has a difference of 1 from a multiple of 7 for all integers 1 ≤ n ≤ 4.
Do show the appropriate multiples of 7
What is disproof by counter-example?
Disproof by counter-examples involves finding a value that does not work for the given statement
That value is called a counter-example
Q: How do you disprove a result?
A: You only need to find one value that does not work.
Q: What types of numbers can have unusual results?
A: Irrational numbers, zero, and certain integers like 2 (the only even prime number) can behave unusually.
Q: Why is zero considered unusual in math?
A: Zero doesn’t always behave like other numbers and can have unexpected results.
Q: Why is 2 considered unusual in terms of prime numbers?
A: 2 is the only even prime number.
Q: What happens when you multiply two numbers?
A: Normally, multiplication gives a larger result than addition, but 2² = 4 looks strange but is true.
Q: What should you check when disproving a result?
A: Look carefully at what types of numbers the result is claimed for, such as integers versus real numbers.
Q: How can irrational numbers be unusual?
A: Irrational numbers can behave strangely in various mathematical contexts.
Use a counter-example to show that the sum of three distinct prime numbers is not always odd:
-All prime numbers are odd, except 2.
-We can expect 2 to be involved.
5 + 11 + 2 = 18.
-Odd + Odd = Even.
-The third number will need to be even.
-18 is not odd.
-The question asks for distinct primes, so all three must be different.
