Unit 2 Logic And Proof Answer Key

Unit 2: Logic and Proof – Answer Key Deep Dive

This comprehensive guide delves into the intricacies of Unit 2: Logic and Proof, providing detailed answers and explanations to common questions and challenges. We’ll explore various aspects of logical reasoning, proof techniques, and problem-solving strategies, ensuring a thorough understanding of this crucial mathematical unit. This isn't just an answer key; it's a learning resource designed to enhance your comprehension and problem-solving skills.

Understanding the Fundamentals of Logic

Before diving into specific problems and their solutions, let's solidify our understanding of the fundamental concepts underpinning logic and proof. This section will serve as a refresher for key terms and definitions.

1. Statements and Truth Values:

A statement is a declarative sentence that is either true or false, but not both. It's crucial to distinguish between statements and non-statements (questions, commands, opinions). The truth value of a statement indicates whether it's true (T) or false (F).

2. Logical Connectives:

Logical connectives link simple statements to form compound statements. The most common are:

• Negation (¬): Reverses the truth value of a statement. For example, if P is "It is raining," ¬P is "It is not raining."
• Conjunction (∧): "And." P ∧ Q is true only if both P and Q are true.
• Disjunction (∨): "Or" (inclusive). P ∨ Q is true if at least one of P or Q is true.
• Conditional (→): "If...then." P → Q is false only if P is true and Q is false.
• Biconditional (↔): "If and only if." P ↔ Q is true if P and Q have the same truth value (both true or both false).

3. Truth Tables:

Truth tables are invaluable tools for analyzing the truth values of compound statements. They systematically list all possible combinations of truth values for the simple statements and determine the corresponding truth value of the compound statement.

4. Logical Equivalence:

Two statements are logically equivalent if they have the same truth value under all possible circumstances. This is often denoted by the symbol ≡. For example, ¬(P ∧ Q) ≡ (¬P ∨ ¬Q) (De Morgan's Law).

Proof Techniques: A Systematic Approach

Mathematical proofs establish the truth of a statement using logical reasoning and previously established facts. Several techniques are commonly employed:

1. Direct Proof:

This approach starts with the given assumptions (premises) and uses logical deductions to arrive at the conclusion. Each step must be justified by a definition, theorem, or previously established result.

2. Indirect Proof (Proof by Contradiction):

This powerful technique assumes the negation of the conclusion and then shows that this assumption leads to a contradiction. Since a contradiction is impossible within a consistent system, the original assumption (negation of the conclusion) must be false, thereby proving the conclusion.

3. Proof by Induction:

Used to prove statements about natural numbers (1, 2, 3,...). This 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 that it's also true for k+1.

Sample Problems and Detailed Solutions

Let's now tackle some sample problems, illustrating the application of the concepts and techniques discussed. Remember, the key is to break down complex problems into smaller, manageable steps.

Problem 1: Construct a truth table for the statement (P ∨ Q) → ¬R.

Problem 2: Prove, using a direct proof, that if n is an even integer, then n² is also an even integer.

Proof: Let n be an even integer. By definition, this means that n can be expressed as n = 2k for some integer k. Now, let's consider n². We have n² = (2k)² = 4k² = 2(2k²). Since 2k² is an integer (because k is an integer), we can see that n² is of the form 2 times an integer. Therefore, by definition, n² is an even integer. This completes the direct proof.

Problem 3: Prove, by contradiction, that √2 is irrational.

Proof: Assume, for the sake of contradiction, that √2 is rational. This means that √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 its simplest form).

Then, we have √2 = a/b. Squaring both sides, we get 2 = a²/b². Rearranging, we get 2b² = a². This implies that a² is an even integer (since it's equal to 2 times another integer). If a² is even, then a must also be even (because the square of an odd number is always odd).

Since a is even, we can write a = 2k for some integer k. Substituting this into 2b² = a², we get 2b² = (2k)² = 4k². Dividing by 2, we get b² = 2k². This means that b² is also even, and therefore b must be even.

But this contradicts our initial assumption that a and b have no common factors, since both a and b are even (and thus have a common factor of 2). This contradiction arises from our assumption that √2 is rational. Therefore, our assumption must be false, and √2 must be irrational.

Problem 4: Prove, by induction, that the sum of the first n natural numbers is given by the formula: 1 + 2 + 3 + ... + n = n(n+1)/2.

Base Case (n=1): The formula gives 1(1+1)/2 = 1, which is the sum of the first natural number.

Inductive Hypothesis: Assume the formula holds for some arbitrary natural number k: 1 + 2 + 3 + ... + k = k(k+1)/2.

Inductive Step: We need to show that the formula also holds for k+1: 1 + 2 + 3 + ... + k + (k+1) = (k+1)(k+2)/2.

Starting with the left-hand side: 1 + 2 + 3 + ... + k + (k+1) = [k(k+1)/2] + (k+1) (using the inductive hypothesis) = [k(k+1) + 2(k+1)]/2 = [(k+1)(k+2)]/2

This is equal to the right-hand side of the formula for k+1. Therefore, by the principle of mathematical induction, the formula holds for all natural numbers n.

Advanced Topics and Further Exploration

This section briefly touches upon more advanced topics within Unit 2: Logic and Proof, encouraging further exploration and deeper understanding.

1. Quantifiers:

Quantifiers specify the scope of a statement. The universal quantifier (∀) means "for all," while the existential quantifier (∃) means "there exists." Understanding how to work with these quantifiers is crucial in more complex logical arguments.

2. Set Theory:

Set theory provides a framework for working with collections of objects. Concepts like subsets, unions, intersections, and power sets are fundamental and often intertwined with logical reasoning.

3. Predicate Logic:

Predicate logic extends propositional logic by incorporating predicates (statements with variables) and quantifiers. This allows for a more nuanced and powerful way of representing and reasoning about statements.

4. Mathematical Structures:

Exploring various mathematical structures like groups, rings, and fields reveals the deeper connections between logic and abstract algebra. The axiomatic approach within these structures demonstrates the power of logical reasoning in building mathematical systems.

This comprehensive guide provides a solid foundation for understanding Unit 2: Logic and Proof. Remember that consistent practice and a thorough grasp of the fundamental concepts are key to mastering this subject. Don't hesitate to revisit challenging problems and explore supplementary resources to strengthen your understanding and build confidence in your problem-solving skills. The journey to mastering logic and proof is a rewarding one, empowering you with valuable critical thinking and reasoning abilities.