Question

Let \(p\) denote the proposition " Jill plays basketball" and \(q\) denote the proposition "Jim plays soccer." Write out-in the clearest way you can-what the following propositions mean: (a) \(\neg p\) (b) \(p \wedge q\) (c) \(p \vee q\) (d) \(\neg p \wedge q\) (c) \(p \rightarrow q\) (f) \(p \leftrightarrow q\) \((g) \neg q \rightarrow p\)

Short answer

(a) Jill does not play basketball; (b) Jill plays basketball and Jim plays soccer; (c) Jill plays basketball or Jim plays soccer; (d) Jill does not play basketball and Jim plays soccer; (e) If Jill plays basketball, then Jim plays soccer; (f) Jill plays basketball if and only if Jim plays soccer; (g) If Jim does not play soccer, then Jill plays basketball.

Step-by-step solution

Understanding the Logic Symbols

We need to understand common logic symbols. The symbol \(eg\) represents "not," \(\wedge\) denotes "and," \(\vee\) stands for "or," \(\rightarrow\) indicates "implies," and \(\leftrightarrow\) means "if and only if." Using these, we will break down each proposition.

Examining Proposition (a)

For the proposition \(eg p\), "Jill plays basketball" is false. Thus, it means "Jill does not play basketball."

Analyzing Proposition (b)

For \(p \wedge q\), both propositions \(p\) (Jill plays basketball) and \(q\) (Jim plays soccer) must be true. Thus, it means "Jill plays basketball and Jim plays soccer."

Exploring Proposition (c)

The symbol \(\vee\) in \(p \vee q\) indicates that at least one of the propositions is true. It translates to "Jill plays basketball or Jim plays soccer."

Handling Proposition (d)

In \(eg p \wedge q\), "Jill does not play basketball" and "Jim plays soccer" both must hold true. Therefore, it means "Jill does not play basketball and Jim plays soccer."

Unpacking Proposition (e)

\(p \rightarrow q\) means "If Jill plays basketball, then Jim plays soccer." It states a conditional dependency where Jill playing basketball implies Jim playing soccer.

Decoding Proposition (f)

The proposition \(p \leftrightarrow q\) translates to an equivalency: "Jill plays basketball if and only if Jim plays soccer," indicating that both propositions are either true or false together.

Interpreting Proposition (g)

For \(eg q \rightarrow p\), it establishes that "If Jim does not play soccer, then Jill plays basketball," meaning Jim's not playing soccer implies Jill playing basketball.

Key concepts

Negation

Negation is a fundamental concept in logic that essentially turns the truth value of a proposition into its opposite. Given a proposition, negation changes true to false and false to true. For instance, if you have a proposition denoted by \( p \) that claims "Jill plays basketball," then the negation of this proposition is expressed as \( eg p \). This reads as "Jill does not play basketball." Think of negation as a logical filter that adjusts the reality of statements by introducing a "not".

• Symbol: \( eg \)
• Operation: Turns truth to falsehood and vice versa
• Example: If \( p \) is true, then \( eg p \) is false

Negation is useful in forming complex logical statements and understanding contradictions within statements.

Conjunction

Conjunction is a logical operator that combines two propositions such that the resulting statement is true only if both propositions are true. Represented by the symbol \( \wedge \), conjunction is quite like the English word "and." For instance, when you encounter a proposition like \( p \wedge q \), it states "Jill plays basketball and Jim plays soccer." Both must actually participate in their respective sports for the conjunction to hold true.

• Symbol: \( \wedge \)
• Requirements: Both propositions must be true
• Example: \( p \wedge q \) is true if both \( p \) and \( q \) are true

This concept helps us understand conditions where multiple scenarios must occur together for a general outcome.

Disjunction

Disjunction offers more flexibility in logic through its use of the "or" operator, denoted by the symbol \( \vee \). In disjunction, the collective proposition is true if at least one of the involved propositions is true. For example, \( p \vee q \) reads "Jill plays basketball or Jim plays soccer," and it remains valid if either Jill plays basketball, Jim plays soccer, or both. Disjunction underlines conditions where achieving at least one requirement is enough for the entire statement to hold.

• Symbol: \( \vee \)
• Requirements: At least one proposition must be true
• Example: \( p \vee q \) is true if at least \( p \) or \( q \) is true

Understanding disjunction is critical for scenarios where options or alternatives are considered.

Implication

Implication reflects a conditional relationship between two propositions, making it an essential tool in logical reasoning. Denoted by \( \rightarrow \), it connects propositions by establishing that the truth of one proposition leads to the truth of another. For instance, \( p \rightarrow q \) translates to "If Jill plays basketball, then Jim plays soccer." Here, Jill’s playing basketball becomes a prerequisite for Jim’s participation in soccer.

• Symbol: \( \rightarrow \)
• Understanding: Truth of the first proposition entails the truth of the second
• Example: \( p \rightarrow q \) is false only when \( p \) is true and \( q \) is false

This concept helps in making inferences and understanding dependencies in logical statements.

Biconditional

The biconditional operator expresses a situation where two propositions are equivalent, meaning that they hold true or false in tandem. Denoted by \( \leftrightarrow \), it stands for "if and only if" (often abbreviated as "iff"). Given a proposition like \( p \leftrightarrow q \), we interpret it as "Jill plays basketball if and only if Jim plays soccer." This indicates that either both situations occur simultaneously, or neither does.

• Symbol: \( \leftrightarrow \)
• Requirements: Both propositions share the same truth value
• Example: \( p \leftrightarrow q \) is true if both \( p \) and \( q \) are either true or false

The biconditional is key in settings where mutual conditions or equivalencies are crucial for logical coherence.