Chapter 1 Flashcards
(38 cards)
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.
