Question

Translate the following expressions into propositional logic. Use the following proposition letters: \(p="\) Jones told the truth." \(q={ }^{*}\) The butler did it." \(r=" I^{\prime} \|\) eat my hat." \(s=\) "The moon is made of green cheese." \(t=\) "If water is heated to \(100^{\circ} \mathrm{C}\), it turns to vapor." (a) "If Jones told the truth. then if the butler did it, I'll eat my hat." (b) "If the butler did it, then either Jones told the truth or the moon is made of green cheese, but not both." (c) "It is not the case that both Jones told the truth and the moon is made of green cheese." (d) "Jones did not tell the truth, and the moon is not made of green cheese, and I'll not eat my hat." (e) "If Jones told the truth implies I'll eat my hat, then if the butler did it, the moon is made of green cheese." (f) "Jones told the truth, and if water is heated to \(100^{\circ} \mathrm{C}\), it turns to vapor."

Short answer

(a) \( p \rightarrow (q \rightarrow r) \), (b) \( q \rightarrow ((p \lor s) \land \neg(p \land s)) \), (c) \( \neg(p \land s) \), (d) \( \neg p \land \neg s \land \neg r \), (e) \( (p \rightarrow r) \rightarrow (q \rightarrow s) \), (f) \( p \land t \).

Step-by-step solution

Understanding the Problem

We need to translate each given English expression into a statement in propositional logic using the proposition letters: \( p \) (Jones told the truth), \( q \) (The butler did it), \( r \) (I'll eat my hat), \( s \) (The moon is made of green cheese), and \( t \) (If water is heated to \(100^{\circ} \mathrm{C}\), it turns to vapor).

Translate Statement (a)

Statement (a): "If Jones told the truth, then if the butler did it, I'll eat my hat." This translates to \( p \rightarrow (q \rightarrow r) \).

Translate Statement (b)

Statement (b): "If the butler did it, then either Jones told the truth or the moon is made of green cheese, but not both." This translates to \( q \rightarrow ((p \lor s) \land eg(p \land s)) \).

Translate Statement (c)

Statement (c): "It is not the case that both Jones told the truth and the moon is made of green cheese." This translates to \( eg(p \land s) \).

Translate Statement (d)

Statement (d): "Jones did not tell the truth, and the moon is not made of green cheese, and I'll not eat my hat." This translates to \( eg p \land eg s \land eg r \).

Translate Statement (e)

Statement (e): "If Jones told the truth implies I'll eat my hat, then if the butler did it, the moon is made of green cheese." This translates to \( (p \rightarrow r) \rightarrow (q \rightarrow s) \).

Translate Statement (f)

Statement (f): "Jones told the truth, and if water is heated to \(100^{\circ} \mathrm{C}\), it turns to vapor." This translates to \( p \land t \).

Key concepts

Logical Translation

Logical translation is the process of converting ordinary language statements into symbolic expressions using propositional logic. This process is essential in mathematics and computer science because it allows us to apply logical reasoning in a precise and unambiguous way. When translating sentences from English to propositional logic, we use specific symbols to represent logical relationships.
Here are some common logical constructs:

• "And" is denoted by \( \land \)
• "Or" is denoted by \( \lor \)
• "Not" is represented by \( eg \)
• "If... then..." is shown by \( \rightarrow \)
• "If and only if" is represented by \( \leftrightarrow \)

Understanding each part of the statement ensures correct translation into logical expressions. Start by identifying the core propositions and logical connectors. Translating complex sentences often requires breaking them into simpler parts that correspond to basic logical operations.

Logical Operators

Logical operators are the symbols or words used to connect statements in propositional logic. Each operator defines a specific type of logical relationship, and understanding these is critical for working with logic statements correctly.
Let's explore some key logical operators:

• **Conjunction (\( \land \))**: This operator represents "and". A statement \( p \land q \) is true only if both \( p \) and \( q \) are true.
• **Disjunction (\( \lor \))**: Representing "or", this is true if at least one of the statements is true. The expression \( p \lor q \) is false only if both \( p \) and \( q \) are false.
• **Negation (\( eg \))**: This operator reflects "not" and inverts the truth value of a statement. If \( p \) is true, \( eg p \) is false, and vice versa.
• **Implication (\( \rightarrow \))**: Reflecting "if... then...", \( p \rightarrow q \) is false only when \( p \) is true and \( q \) is false.
• **Biconditional (\( \leftrightarrow \))**: This operator shows "if and only if", meaning \( p \leftrightarrow q \) is true if both \( p \) and \( q \) are either true or false at the same time.

Using these operators helps to form precise logical statements essential for rigorous reasoning and problem-solving in various fields.

Discrete Mathematics

Discrete mathematics is a branch of mathematics focused on countable, distinct elements. It includes a variety of topics such as logic, set theory, combinatorics, graph theory, and discrete probability. In particular, propositional logic is a fundamental area within discrete mathematics.

Importance of Discrete Mathematics

Discrete mathematics lays the foundation for the logical structures used in computer science. Algorithms, data structures, and programming rely heavily on concepts from discrete mathematics.

• **Logic**: Propositional logic is a stepping stone for understanding more complex logical systems utilized in computer algorithms.
• **Structure**: It offers a framework to understand complex relationships through clearly defined rules.
• **Efficiency**: Simplifies complex problems by providing methods for precise calculations and proofs.

In essence, discrete mathematics allows for the analysis and description of phenomena where continuity does not play a role, fostering the development of computational techniques.

Propositional Variables

Propositional variables serve as placeholders for sentences or propositions that can either be true or false. These variables are typically represented by letters like \( p, q, r, s, \) and \( t \), and form the basis for expressions in propositional logic.

Understanding Propositional Variables

Propositional variables are simple yet powerful tools in logic:

• They allow you to model complex logical expressions succinctly.
• Using variables, you can form logical expressions that capture the essence of entire sentences or ideas.
• These variables are then manipulated with logical operators to test various conditions and outcomes.
• In conditional statements, they serve to simplify the process of determining the overall truth value of a complex sentence by abstracting specific scenarios.

Propositional variables are fundamental to logic and mathematics as they provide a straightforward, symbolic way to express statements and explore their logical consequences.