Problem 43 - Discrete Mathematics with Applications

Question

Let \(a, b,\) and \(c\) be any real numbers. Then \(a

Step-by-Step Solution

Answer

Using the logical expression \(p\rightarrow (q \vee r) = (p \wedge \sim q) \rightarrow r\), we defined variables \(p\) as "\(a\cdot b = 0\)", \(q\) as "\(a = 0\)", and \(r\) as "\(b = 0\)". We assumed \((p \wedge \sim q)\) to be true, meaning that \(a\cdot b = 0\) and \(a \neq 0\). Utilizing the given fact, we determined that since \(a\cdot b=0\) and \(a\neq 0\), dividing both sides by \(a\) gives \(b = 0\). Thus, for any real numbers \(a, b\) if \(a\cdot b = 0\), then either \(a = 0\) or \(b = 0\).

Step 1: Define variables as the statements in the given hint

Let \(p\) be the statement "\(a\cdot b=0\)", \(q\) be the statement "\(a=0\)", and \(r\) be the statement "\(b=0\)". We need to prove that \(p\rightarrow (q \vee r)\). Using the hint, we rewrite this as \((p \wedge \sim q)\rightarrow r\).

Step 2: Assume \((p \wedge \sim q)\) to be true

Since we are trying to prove \((p \wedge \sim q) \rightarrow r\), we will assume \((p \wedge \sim q)\) is true for this step. This means that both \(a\cdot b = 0\) and \(a \neq 0\) are true. Our goal now is to show that \(b=0\).

Step 3: Utilize the given fact about real numbers

We are given the fact that: \(a

Step 4: Show that \(b=0\)

Since \(a\cdot b=0\) and \(a\neq 0\), we can divide both sides of the equation by \(a\) to isolate \(b\): \(b = \frac{0}{a} \) As \(0\) divided by any non-zero number equals \(0\), we have: \(b = 0\) So, under the assumption of \((p \wedge \sim q)\), we have proven that \(b=0\) which means \(r\) is true.

Step 5: Conclude the result

We have shown that \((p \wedge \sim q) \rightarrow r\) is true, so using the hint, we have \(p\rightarrow (q \vee r)\), which means that for any real numbers \(a, b\) if \(a\cdot b = 0\), then either \(a=0\) or \(b=0\).

Key Concepts

Real Numbers

Real numbers form the backbone of mathematics. They include all the numbers on the number line: positive numbers, negative numbers, and zero. They are used extensively in various math operations and provide a complete set of numbers for count, measure, and calculation. Here are some categories within real numbers:

• Natural numbers: Counting numbers starting from 1, 2, 3, ..., etc.
• Whole numbers: Natural numbers including zero.
• Integers: Whole numbers including negative counterparts like -3, -2, -1, 0, 1, 2, 3, ...
• Rational numbers: Numbers expressed as a ratio of two integers, e.g., \(\frac{3}{4}\).
• Irrational numbers: Numbers that cannot be expressed as a simple fraction, like \(\pi\) or \(\sqrt{2}\).

Real numbers are crucial because they allow us to perform a wide range of mathematical operations effectively. They also help define various properties, such as inequalities, which are often used to formulate logical proofs. In the context of proofs, real numbers maintain certain properties such as order and completeness, which can be leveraged to demonstrate specific mathematical assertions.

Logical Statements

In mathematics, logical statements are expressions that can be either true or false. The truth value of these statements is an essential part of constructing mathematical proofs. Logical statements help to establish connections or relationships between different mathematical conditions or equations.

• Basic Elements: Terms like "and" (\(\land\)), "or" (\(\lor\)), and "not" (\(\lnot\)) are commonly used.
• Examples: "\(a = 0\)" is a simple statement. "\(a eq 0\)" uses negation.
• Compound Statements: Combined using connectors like "and" or "or". Example: "\((a = 0) \lor (b = 0)\)" means "either \(a\) is 0 or \(b\) is 0".

Logical thinking and the use of structured statements are key to forming arguments and proving or disproving mathematical hypotheses. Recognizing how to manipulate and interpret these statements is instrumental in problem-solving across countless areas of math.

Implication in Logic

Implication is a fundamental concept in logic represented by "\(\rightarrow\)". It captures the notion of one statement leading to another. For instance, if one statement \(p\) implies another statement \(q\), written \(p \rightarrow q\), it means if \(p\) is true, then \(q\) must also be true.

• Basic Example: If it rains (\(p\)), then the ground is wet (\(q\)).
• Structure: Implications are often used to establish conditions or necessary outcomes.
• Logical Equivalences: Such as "if \(p\) then \(q\)" is logically equivalent to "\(\lnot p \lor q\)".

Understanding implication is key in proofs. For instance, the exercise shows how making assumptions (like "\((p \land \lnot q)\) implies \(r\)") helps reach conclusions ("\(b = 0\)"). It's a powerful tool for what logically follows from given facts and assumptions, enabling indirect and direct evidential claims.

Distributive Property

The distributive property is a cornerstone of algebra that allows us to simplify and solve equations by distributing one operation over another. It states that for all real numbers \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equal to \(ab + ac\). This property is useful in a variety of mathematical scenarios, including simplifying expressions and solving equations.

• Formula: \(a(b + c) = ab + ac\)
• Application: Simplifies complex expressions into manageable terms.
• Use in Proofs: Helps rearrange and redefine expressions to reach valid conclusions.

In the provided proof exercise, understanding and utilizing properties like distribution can help guide logical deductions and foster a deeper comprehension of mathematical relationships. Though the distributive property won't be directly involved in this exercise, knowing how to manipulate expressions is invaluable when formulating logical evidence and reasoning.

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Problem 43

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Problem 43

Other exercises in this chapter

Problem 42

Use De Morgan's laws to verify each. (Hint: \(p \rightarrow q \equiv \sim p \vee q\) ). $$\sim(\sim p \wedge \sim q) \equiv p \vee q$$

View solution

Problem 43

Use De Morgan's laws to verify each. (Hint: \(p \rightarrow q \equiv \sim p \vee q\) ). $$\sim(\sim p \vee q) \equiv p \wedge \sim q$$

View solution

Problem 43

Determine whether or not the assignment statement \(x \leftarrow x+1\) will be executed in each sequence of statements, where \(i \leftarrow 2, j \leftarrow 3,

View solution

Problem 44

Use De Morgan's laws to verify each. (Hint: \(p \rightarrow q \equiv \sim p \vee q\) ). $$\sim(\sim p \vee \sim q) \equiv p \wedge q$$

View solution

Key Concepts

Real Numbers

Real numbers form the backbone of mathematics. They include all the numbers on the number line: positive numbers, negative numbers, and zero. They are used extensively in various math operations and provide a complete set of numbers for count, measure, and calculation. Here are some categories within real numbers:

• Natural numbers: Counting numbers starting from 1, 2, 3, ..., etc.
• Whole numbers: Natural numbers including zero.
• Integers: Whole numbers including negative counterparts like -3, -2, -1, 0, 1, 2, 3, ...
• Rational numbers: Numbers expressed as a ratio of two integers, e.g., \(\frac{3}{4}\).
• Irrational numbers: Numbers that cannot be expressed as a simple fraction, like \(\pi\) or \(\sqrt{2}\).

Real numbers are crucial because they allow us to perform a wide range of mathematical operations effectively. They also help define various properties, such as inequalities, which are often used to formulate logical proofs. In the context of proofs, real numbers maintain certain properties such as order and completeness, which can be leveraged to demonstrate specific mathematical assertions.

Logical Statements

In mathematics, logical statements are expressions that can be either true or false. The truth value of these statements is an essential part of constructing mathematical proofs. Logical statements help to establish connections or relationships between different mathematical conditions or equations.

• Basic Elements: Terms like "and" (\(\land\)), "or" (\(\lor\)), and "not" (\(\lnot\)) are commonly used.
• Examples: "\(a = 0\)" is a simple statement. "\(a eq 0\)" uses negation.
• Compound Statements: Combined using connectors like "and" or "or". Example: "\((a = 0) \lor (b = 0)\)" means "either \(a\) is 0 or \(b\) is 0".

Logical thinking and the use of structured statements are key to forming arguments and proving or disproving mathematical hypotheses. Recognizing how to manipulate and interpret these statements is instrumental in problem-solving across countless areas of math.

Implication in Logic

Implication is a fundamental concept in logic represented by "\(\rightarrow\)". It captures the notion of one statement leading to another. For instance, if one statement \(p\) implies another statement \(q\), written \(p \rightarrow q\), it means if \(p\) is true, then \(q\) must also be true.

• Basic Example: If it rains (\(p\)), then the ground is wet (\(q\)).
• Structure: Implications are often used to establish conditions or necessary outcomes.
• Logical Equivalences: Such as "if \(p\) then \(q\)" is logically equivalent to "\(\lnot p \lor q\)".

Understanding implication is key in proofs. For instance, the exercise shows how making assumptions (like "\((p \land \lnot q)\) implies \(r\)") helps reach conclusions ("\(b = 0\)"). It's a powerful tool for what logically follows from given facts and assumptions, enabling indirect and direct evidential claims.

Distributive Property

The distributive property is a cornerstone of algebra that allows us to simplify and solve equations by distributing one operation over another. It states that for all real numbers \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equal to \(ab + ac\). This property is useful in a variety of mathematical scenarios, including simplifying expressions and solving equations.

• Formula: \(a(b + c) = ab + ac\)
• Application: Simplifies complex expressions into manageable terms.
• Use in Proofs: Helps rearrange and redefine expressions to reach valid conclusions.

In the provided proof exercise, understanding and utilizing properties like distribution can help guide logical deductions and foster a deeper comprehension of mathematical relationships. Though the distributive property won't be directly involved in this exercise, knowing how to manipulate expressions is invaluable when formulating logical evidence and reasoning.