Problem 38 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The short answer based on the step-by-step solution is: The sentence "If \(x \geq y\) and \(x< z,\) then \(y< z\)" can be represented symbolically as \[(\neg p \land r) \rightarrow q\]

Step 1: Identification

The given symbolic representations are: - \(p: x

Step 2: Propositions

The propositions involved in the sentence are: - \(x \geq y\): This is the negation of the given proposition \(p\). So, we can represent this as \(\neg p\). - \(x < z\): This is given by the proposition \(r\). - \(y < z\): This is given by the proposition \(q\). Write down the compound proposition in the antecedent.

Step 3: Antecedent

The antecedent has two parts: 1. \(x \geq y\) which is \(\neg p\) 2. \(x < z\) which is \(r\) Since both parts need to be true in the antecedent, we need to use the conjunction symbol \((\land)\). So, the antecedent can be written as \((\neg p \land r)\). Combine the antecedent and consequent using the implication symbol.

Step 4: Conditional statement

Now, we know the antecedent \((\neg p \land r)\) and the consequent \(q\). To represent the entire sentence symbolically, we can use the implication symbol \((\rightarrow)\). The symbolic representation of the given sentence is: \[(\neg p \land r) \rightarrow q\]

Key Concepts

Propositional Variables

In symbolic logic, we often use propositional variables to represent statements or expressions that can either be true or false. This helps us simplify complex logical expressions. For instance, in the given exercise, we are handling three propositional variables:

• q: The statement "y < z," where y is less than z.

• r: The statement "x < z," where x is less than z.

By using these propositional variables, we can interact with logical structures more easily. Each variable here is a simple statement that lays the groundwork for constructing more complex logical expressions.

Logical Conjunction

The concept of logical conjunction is crucial in symbolic logic. Conjunction refers to the combination of two propositions using the "and" operator, typically symbolized by \( \land \). In logical terms, a conjunction of two statements is true only if both individual statements are true.

For example, in our exercise, we encounter two statements:

• "\( x \geq y \)" is represented by \( eg p \) because it is the negation of "x < y."
• "x < z," which directly reflects \( r \).

To express the simultaneous truth of these conditions, we use logical conjunction. The formula \( eg p \land r \) symbolizes that both conditions must be true together.

Logical Implication

Logical implication is a foundational concept in formal logic. It denotes a conditional relationship between two propositions and is symbolized by \( \rightarrow \). An implication \( (A \rightarrow B) \) means that if A is true, then B must also be true.

In our exercise, we have a compound statement involving implication: \((eg p \land r) \rightarrow q\). This means that if \( eg p \land r \) is true, then q must also be true.

• The antecedent (what comes before \( \rightarrow \)) is \( eg p \land r \), meaning "\( x \geq y \) and \( x < z \)."
• The consequent (what comes after \( \rightarrow \)) is \( q \), meaning "y < z."

Thus, the logical implication here expresses a relationship that aligns with traditional "if...then" logic in mathematics and reasoning.

Negation in Logic

Negation in logic is the operation that takes a proposition and inverts its truth value. When we state the negation of a proposition, we are claiming that the opposite of what is stated is true. It is typically denoted by the symbol \( eg \).

In the original exercise, the statement "\( x \geq y \)" is the negation of the proposition "x < y," which is represented as \( p \). To represent this logically, we write \( eg p \), meaning it is not true that "x < y," or equivalently, "x is greater than or equal to y."

Negation helps us express conditions that exclude a particular state of affairs, allowing us to modify and understand logical statements better.

Previous

Problem 38

Next

Problem 39

Other exercises in this chapter

Problem 37

Write the converse, inverse, and contrapositive of each implication. If London is in France, then Paris is in England.

View solution

Problem 38

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 \equiv q

View solution

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

View solution

Problem 39

Find the flaw in the following "proof": Let \(a\) and \(b\) be real numbers such that \(a=b .\) Then \(a b=b^{2}\) Therefore, \(a^{2}-a b=a^{2}-b^{2}\) Factorin

View solution

Key Concepts

Propositional Variables

In symbolic logic, we often use propositional variables to represent statements or expressions that can either be true or false. This helps us simplify complex logical expressions. For instance, in the given exercise, we are handling three propositional variables:

• q: The statement "y < z," where y is less than z.

• r: The statement "x < z," where x is less than z.

By using these propositional variables, we can interact with logical structures more easily. Each variable here is a simple statement that lays the groundwork for constructing more complex logical expressions.

Logical Conjunction

The concept of logical conjunction is crucial in symbolic logic. Conjunction refers to the combination of two propositions using the "and" operator, typically symbolized by \( \land \). In logical terms, a conjunction of two statements is true only if both individual statements are true.

For example, in our exercise, we encounter two statements:

• "\( x \geq y \)" is represented by \( eg p \) because it is the negation of "x < y."
• "x < z," which directly reflects \( r \).

To express the simultaneous truth of these conditions, we use logical conjunction. The formula \( eg p \land r \) symbolizes that both conditions must be true together.

Logical Implication

Logical implication is a foundational concept in formal logic. It denotes a conditional relationship between two propositions and is symbolized by \( \rightarrow \). An implication \( (A \rightarrow B) \) means that if A is true, then B must also be true.

In our exercise, we have a compound statement involving implication: \((eg p \land r) \rightarrow q\). This means that if \( eg p \land r \) is true, then q must also be true.

• The antecedent (what comes before \( \rightarrow \)) is \( eg p \land r \), meaning "\( x \geq y \) and \( x < z \)."
• The consequent (what comes after \( \rightarrow \)) is \( q \), meaning "y < z."

Thus, the logical implication here expresses a relationship that aligns with traditional "if...then" logic in mathematics and reasoning.

Negation in Logic

Negation in logic is the operation that takes a proposition and inverts its truth value. When we state the negation of a proposition, we are claiming that the opposite of what is stated is true. It is typically denoted by the symbol \( eg \).

In the original exercise, the statement "\( x \geq y \)" is the negation of the proposition "x < y," which is represented as \( p \). To represent this logically, we write \( eg p \), meaning it is not true that "x < y," or equivalently, "x is greater than or equal to y."

Negation helps us express conditions that exclude a particular state of affairs, allowing us to modify and understand logical statements better.