2-4 Practice Writing Proofs Answer Key

2-4 Practice Writing Proofs: Answer Key & Comprehensive Guide

This comprehensive guide provides detailed solutions and explanations for common 2-4 practice writing proofs exercises, equipping you with the skills and understanding needed to confidently tackle more complex mathematical proofs. We'll explore various proof techniques, common pitfalls to avoid, and strategies for improving your proof-writing abilities.

Understanding the Fundamentals of Mathematical Proofs

Before diving into specific examples, let's solidify our understanding of what constitutes a valid mathematical proof. A proof is a logical argument demonstrating the truth of a statement, called a theorem or proposition. This argument must proceed from established axioms, definitions, and previously proven theorems, using logical inferences. Key characteristics of a good proof include:

• Clarity and Precision: Each step must be clearly stated and justified. Ambiguity should be avoided.
• Logical Flow: The argument should progress logically from one step to the next, with each step following directly from the previous one.
• Completeness: All necessary steps must be included. No assumptions should be left implicit.
• Correctness: Every step must be logically sound and based on valid mathematical principles.

Common Proof Techniques

Several techniques are frequently used in mathematical proofs. Mastering these is crucial for success:

1. Direct Proof

This is the most straightforward approach. We start with the given premises (hypotheses) and, through a series of logical deductions, arrive at the conclusion.

Example: Prove that if x is an even integer, then x² is an even integer.

Proof: Let x be an even integer. By definition, this means x = 2k for some integer k. Then, x² = (2k)² = 4k² = 2(2k²). Since 2k² is an integer, x² is of the form 2 times an integer, thus x² is even. ∎ (The symbol ∎ denotes the end of a proof.)

2. Proof by Contradiction (Reductio ad Absurdum)

This method assumes the negation of the conclusion and shows that this leads to a contradiction. Since a contradiction implies the original assumption is false, the conclusion must be true.

Example: Prove that √2 is irrational.

Proof: Assume, for the sake of contradiction, that √2 is rational. Then √2 can be expressed as a fraction a/b, where a and b are integers, b ≠ 0, and a and b have no common factors (the fraction is in lowest terms). Squaring both sides, we get 2 = a²/b², which implies 2b² = a². This means a² is even, and therefore a must be even (as shown in the direct proof example above). We can write a = 2k for some integer k. Substituting this into 2b² = a², we get 2*b² = (2k)² = 4k², which simplifies to b² = 2k². This means b² is even, and therefore b is even. But this contradicts our initial assumption that a and b have no common factors (since both are even). This contradiction means our initial assumption (that √2 is rational) must be false. Therefore, √2 is irrational. ∎

3. Proof by Induction

This technique is particularly useful for proving statements about natural numbers (1, 2, 3, ...). It involves two steps:

• Base Case: Prove the statement is true for the smallest natural number (usually 1).
• Inductive Step: Assume the statement is true for some arbitrary natural number k (the inductive hypothesis), and then prove it's also true for k+1.

Example: Prove that the sum of the first n natural numbers is n(n+1)/2.

Proof:

• Base Case (n=1): The sum of the first 1 natural number is 1, and 1(1+1)/2 = 1. The statement holds for n=1.
• Inductive Step: Assume the statement is true for some arbitrary k: 1 + 2 + ... + k = k(k+1)/2. We need to show it's true for k+1: 1 + 2 + ... + k + (k+1) = [k(k+1)/2] + (k+1) = [*k(k+1) + 2(k+1)]/2 = [(k+1)(k+2)]/2 = (k+1)(k+1+1)/2.

The statement holds for k+1. By the principle of mathematical induction, the statement is true for all natural numbers n. ∎

2-4 Practice Writing Proofs: Sample Problems & Solutions

Let's tackle some practice problems, illustrating the application of these techniques. Remember, the key is to break down the problem into manageable steps, clearly justifying each transition.

Problem 1: Prove that the sum of two even integers is an even integer.

Solution: Let a and b be two even integers. By definition, a = 2m and b = 2n for some integers m and n. Their sum is a + b = 2m + 2n = 2(m + n). Since m + n is an integer, a + b is of the form 2 times an integer, hence it's even. ∎

Problem 2: Prove that if n is an integer and n³ is even, then n is even.

Solution: We'll use proof by contradiction. Assume n is odd. Then n = 2k + 1 for some integer k. Then n³ = (2k + 1)³ = 8k³ + 12k² + 6k + 1 = 2(4k³ + 6k² + 3k) + 1. This shows that n³ is odd, contradicting our given premise that n³ is even. Therefore, our assumption that n is odd must be false, and n must be even. ∎

Problem 3: Prove that for all integers n ≥ 1, 1³ + 2³ + ... + *n³ = [n(n+1)/2]².

Solution: We use proof by induction.

• Base Case (n=1): 1³ = 1, and [1(1+1)/2]² = 1. The statement holds for n=1.
• Inductive Step: Assume the statement is true for some arbitrary k: 1³ + 2³ + ... + *k³ = [k(k+1)/2]². We need to show it's true for k+1: 1³ + 2³ + ... + *k³ + (k+1)³ = [k(k+1)/2]² + (k+1)³ = (*k+1)²[*k²/4 + (k+1)] = (*k+1)²[*k² + 4k + 4]/4 = (*k+1)²(k+2)²/4 = [(k+1)(k+2)/2]². This is the statement for n = k+1. By the principle of mathematical induction, the statement is true for all integers n ≥ 1. ∎

Problem 4: Prove that if x and y are integers such that x + y is even, then x and y are both even or both odd.

Solution: We'll use a proof by cases.

• Case 1: x is even. Then x = 2k for some integer k. Since x + y is even, let x + y = 2m for some integer m. Then y = 2m - x = 2m - 2k = 2(m - k). Since m - k is an integer, y is even.
• Case 2: x is odd. Then x = 2k + 1 for some integer k. Since x + y is even, x + y = 2m. Then y = 2m - x = 2m - (2k + 1) = 2(m - k) - 1. Since m - k is an integer, y is odd.

Therefore, if x + y is even, then x and y are either both even or both odd. ∎

Improving Your Proof-Writing Skills

Practice is key. The more proofs you write, the better you'll become at identifying the appropriate techniques and constructing logical arguments. Here are some tips:

• Start with simple problems: Build a strong foundation before tackling more challenging proofs.
• Understand the definitions: Make sure you thoroughly understand the definitions of all terms involved.
• Break down the problem: Divide the problem into smaller, manageable steps.
• Write clearly and concisely: Use precise language and avoid ambiguity.
• Check your work: Review your proof carefully, ensuring each step is justified and logically sound.
• Seek feedback: Ask others to review your proofs and provide constructive criticism.

By consistently practicing and applying these strategies, you can significantly improve your proof-writing abilities, leading to greater success in mathematics and related fields. Remember, mastering mathematical proofs is a journey that requires patience, persistence, and a keen eye for detail. The rewards, however, are well worth the effort.