Problem 39 - Discrete Mathematics with Applications

Question

Let \(x, y,\) and \(z\) be any real numbers. Represent each sentence symbolically, where \(p: xx\).

Step-by-Step Solution

Answer

The given sentence "If z ≥ y and x < y, then z > x" can be represented symbolically using the statements p, q, and r as: (z≥y) ∧ p → r.

Step 1: Define the given statements as inequalities

We can represent each statement symbolically as follows: p: x < y q: y < z r: x < z We can now use these statements to represent the given sentence, "If z ≥ y and x < y, then z > x."

Step 2: Rewrite the given sentence using logical statements

We can represent the given sentence "If z ≥ y and x < y, then z > x" as an implication with two premises (z ≥ y, and x < y) and one conclusion (z > x). Symbolically, this can be represented as ((z≥y) ∧ (xx).

Step 3: Represent the given sentence using the symbolic statements p, q, and r

Now let's rewrite the sentence using the symbolic statements p, q, and r. Since we have (x

Key Concepts

Inequalities

Inequalities are mathematical expressions showing the relationship between two values, often involving comparisons like greater than, less than, or equal to. In the context of this exercise, inequalities are used to compare real numbers like \(x, y,\) and \(z\). For example, if \(p: x < y\), it clearly indicates that the value of \(x\) is less than the value of \(y\).
The inequalities help us set the stage for our logical statements by creating clear relational boundaries.

• "<" implies that one number is smaller than the other.
• "\(\geq\)" means one number is greater than or equal to another.
• Combining inequalities allows us to deduce further relationships between numbers.

Understanding these simple relationships is crucial in transforming verbal conditions into mathematical expressions that can be evaluated logically.

Logical Statements

Logical statements are composed of declarative sentences dealing with truth values. In this exercise, we employ symbolic logic to transform conditions into mathematical expressions so we can analyze them systematically.
A logical statement takes simple assertions and uses connective words like "and", "or", "if", "then" to form complex expressions.

• "\(\land\)" (AND) - Both conditions must be true.
• "\(\lor\)" (OR) - At least one of the conditions is true.
• Logical operators help us relate different statements to draw new conclusions.

By placing inequalities in the framework of logical statements, we can set the foundation for evaluating them through implication, aiding in the verification of our conditions.

Implication

The concept of implication in logic is similar to "if...then" statements in everyday language. It's a conditional term used in forming logical statements. In this exercise, the implication is crafted by combining the given conditions with the desired result as a logical consequence.
To understand implication:

• If the conditions \((z \geq y)\) and \(x < y\) are true, then \(z > x\) should follow as true for the overall statement to be valid.
• Symbolically, this is represented as \(((z \geq y) \land (x < y)) \rightarrow (z > x)\).
• The implication suggests a causal relationship between the premises and conclusion.

A correct implication confirms that the presumed relationships hold under specified conditions, allowing us to draw reliable broader conclusions about inequalities and logical relationships among variables.

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Problem 39

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Problem 40

Other exercises in this chapter

Problem 39

Mark each sentence as true or false, where \(p, q,\) and \(r\) are arbitrary statements, \(t\) a tautology, and \(f\) a contradiction. If \(p \equiv q\) and \(q

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Problem 39

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Problem 40

Mark each sentence as true or false, where \(p, q,\) and \(r\) are arbitrary statements, \(t\) a tautology, and \(f\) a contradiction. $$\text { If } p \wedge q

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Problem 40

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

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Key Concepts

Inequalities

Inequalities are mathematical expressions showing the relationship between two values, often involving comparisons like greater than, less than, or equal to. In the context of this exercise, inequalities are used to compare real numbers like \(x, y,\) and \(z\). For example, if \(p: x < y\), it clearly indicates that the value of \(x\) is less than the value of \(y\).
The inequalities help us set the stage for our logical statements by creating clear relational boundaries.

• "<" implies that one number is smaller than the other.
• "\(\geq\)" means one number is greater than or equal to another.
• Combining inequalities allows us to deduce further relationships between numbers.

Understanding these simple relationships is crucial in transforming verbal conditions into mathematical expressions that can be evaluated logically.

Logical Statements

Logical statements are composed of declarative sentences dealing with truth values. In this exercise, we employ symbolic logic to transform conditions into mathematical expressions so we can analyze them systematically.
A logical statement takes simple assertions and uses connective words like "and", "or", "if", "then" to form complex expressions.

• "\(\land\)" (AND) - Both conditions must be true.
• "\(\lor\)" (OR) - At least one of the conditions is true.
• Logical operators help us relate different statements to draw new conclusions.

By placing inequalities in the framework of logical statements, we can set the foundation for evaluating them through implication, aiding in the verification of our conditions.

Implication

The concept of implication in logic is similar to "if...then" statements in everyday language. It's a conditional term used in forming logical statements. In this exercise, the implication is crafted by combining the given conditions with the desired result as a logical consequence.
To understand implication:

• If the conditions \((z \geq y)\) and \(x < y\) are true, then \(z > x\) should follow as true for the overall statement to be valid.
• Symbolically, this is represented as \(((z \geq y) \land (x < y)) \rightarrow (z > x)\).
• The implication suggests a causal relationship between the premises and conclusion.

A correct implication confirms that the presumed relationships hold under specified conditions, allowing us to draw reliable broader conclusions about inequalities and logical relationships among variables.