Proof Techniques

In mathematics, especially when studying logic, reasoning and proofs, we need methods to prove that certain mathematical statements are true. These methods or techniques form the backbone of mathematical reasoning. They help us verify the validity of our assumptions and reach logical conclusions based on these assumptions.

What is the evidence?

A proof is a logical series of statements or steps, each of which is reasonably true, that leads to the conclusion that a particular mathematical statement is true. It serves as a concrete demonstration that the conclusion follows from the premises based on the rules of logical deduction.

Proof techniques in mathematics

Several standard proof techniques are widely used in mathematics. Each technique serves a specific purpose and is best suited for particular types of problems or statements. Below we will explore some of the fundamental proof techniques used in grade 11 mathematics, including visual and textual examples to enhance understanding.

1. Direct evidence

Direct proof is a direct way to prove a statement, in which we start with known facts or assumptions and use logical steps to arrive at the statement we want to prove. It is typically used when we need to demonstrate the truth of an implication P → Q

Example

Let us prove the statement: "If a number is even, then its square will also be even."

Let n be an even number. By definition of even numbers, there exists an integer k such that n = 2k. Therefore, n² = (2k)² = 4k². Since 4k² = 2(2k²), n² is divisible by 2, which means n² is even. Hence, if n is even, then n² is also even.

2. Indirect proof (proof by contradiction)

In indirect proof or proof by contradiction, we assume that the statement we want to prove is false. We then show that this assumption leads to a contradiction with known facts, which means that our original statement must be true.

Example

Prove that √2 is irrational.

Assume the opposite, that √2 is rational. Then, by definition of rational numbers, √2 = a/b where a and b are integers, with b ≠ 0 and a/b in simplest form. Thus, 2 = a²/b², or a² = 2b². This implies a² is even, so a is even (since squares of odd numbers are odd). Let a = 2k, so a² = (2k)² = 4k² = 2b². Simplifying gives b² = 2k², showing b² is even, so b is even. Both a and b being even contradicts a/b being in simplest form. Thus, our assumption is wrong, and √2 is irrational.

3. Proof by contrapositive

If we have an implication P → Q, then the contrapositive proof consists in proving ¬Q → ¬P. If ¬Q → ¬P is valid, then P → Q must also be valid.

Example

Prove: "If the square of a number n is odd, then n is also odd."

We consider the contrapositive: "If n is even, then n² is even." Assume that n is even, meaning n = 2k for some integer k. Then n² = (2k)² = 4k² = 2(2k²). Since n² is a multiple of 2, it must be even. This proves the contrapositive, so the original statement is true.

4. Proof by mathematical induction

Induction is a powerful proof technique that is commonly used to prove statements about integers. It involves two main steps: the base case and the inductive step.

• Base case: Verify that the statement is true for the initial value.
• Inductive step: Assume the statement is true for any arbitrary integer k, and prove it for k + 1.

Example

Prove that the sum of the first n positive integers is S(n) = n(n + 1)/2.

Base case: For n = 1, S(1) = 1(1+1)/2 = 1. The statement holds. Inductive step: Assume S(k) = k(k + 1)/2 is true for some k. We need to prove that S(k + 1) = (k + 1)((k + 1) + 1)/2. Consider S(k + 1) = S(k) + (k + 1). By the inductive hypothesis: S(k) = k(k + 1)/2 So, S(k + 1) = k(k + 1)/2 + (k + 1) = (k(k + 1) + 2(k + 1))/2 = (k + 1)(k + 2)/2 The statement holds for k + 1, so by induction, it is true for all n.

5. Proof by exhaustion (case analysis)

This method consists in dividing the statement into a finite number of cases and proving each one separately. It is useful when the statement or its variables allow such division.

Example

Prove that when a fair dice is thrown, it always shows a face less than 7.

The faces of a standard dice display the numbers 1, 2, 3, 4, 5, and 6. We can consider each case:

• Case 1: The dice shows 1 (1 < 7).
• Case 2: The dice shows 2 (2 < 7).
• Case 3: The dice shows 3 (3 < 7).
• Case 4: The dice shows 4 (4 < 7).
• Case 5: The dice shows 5 (5 < 7).
• Case 6: The dice shows 6 (6 < 7).

Since each possible outcome is less than 7, the statement is true.

6. Iterative proof

Although there is not always a standard classification, iterative proof involves using iterative reasoning to establish that a repetitive process gives a particular result. It is often used in algorithms or recursive functions.

Example

Consider the Fibonacci sequence, where each number is the sum of the two preceding numbers: F(0) = 0, F(1) = 1, and F(n) = F(n-1) + F(n-2) where n ≥ 2.

Prove: The number of ways to climb a staircase with n steps, 1 or 2 steps at a time, is the nth Fibonacci number.

Base cases: F(0) = 0 implies no steps to climb = 1 way (stay put). F(1) = 1 implies one step to climb = 1 way. For n ≥ 2, assume the number of ways corresponds to Fibonacci sequence: F(n) = F(n-1) + F(n-2). This matches taking one step to reach (n-1) followed by another step, or two steps to reach (n-2). By induction and iteration through defined conditions, each follows the Fibonacci relationship.

Visual representation

Let's look at a simple visual illustration of mathematical induction using the domino effect:

Think of each domino as representing a case of mathematical induction. Falling the first domino (the base case) ensures that all subsequent dominoes (the cases) fall, this is similar to demonstrating the truth of a statement for all natural numbers by induction.

Conclusion

Proof techniques are essential for understanding and achieving mathematical maturity. They provide a rigorous framework for establishing hypotheses and logically deducing conclusions from given premises. Direct proof, indirect proof, proof by counterpositive, proof by mathematical induction, and proof by exhaustion are the primary methods covered in the grade 11 math curriculum, and they lay the foundation for more advanced mathematical reasoning.

Different types of problems require different proof strategies, and mastering these techniques is vital to progressing in mathematics. As you practice and become familiar with these methods, your ability to think logically and solve complex problems will greatly increase.

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