Problem 30 - Thinking Mathematically

Question

Let \(p\) and \(q\) represent the following simple statements: \(p\) : The campus is closed. q: It is Sunday. Write each compound statement in symbolic form. The campus is not closed if and only if it is not Sunday.

Step-by-Step Solution

Answer

'The campus is not closed if and only if it is not Sunday.' is translated to '~p \leftrightarrow ~q' in symbolic logic.

Step 1: Defining the Statements

First, we assign each simple statement to a variable. \(p\) is assigned to 'The campus is closed.' and \(q\) is assigned to 'It is Sunday.'.

Step 2: Identifying Logical Connectives

We identify the logical connectives used in the compound statement 'The campus is not closed if and only if it is not Sunday.' The logical connective 'not' changes the truth value of a statement; it is represented in symbolic logic as '~'. The logical connective 'if and only if' represents a biconditional, where both implications must be true; it is represented in symbolic logic as '\leftrightarrow'.

Step 3: Translating the Compound Statement

We now translate the entire statement into its symbolic form. The negation of \(p\) (the campus is not closed) is written as '~p'. The negation of \(q\) (it is not Sunday) is written as '~q'. The entire compound statement 'The campus is not closed if and only if it is not Sunday.' can therefore be translated to '~p \leftrightarrow ~q'.

Key Concepts

Biconditional

Understanding the biconditional logical connective is crucial for unraveling the essence of certain compound statements. In the context of symbolic logic, a biconditional relationship between two statements, exemplified by 'p' and 'q', can be identified by the phrase 'if and only if'. This phrase indicates that 'p' is true precisely when 'q' is true, and vice versa.

In mathematical notation, the biconditional is represented by \( \leftrightarrow \) and produces a compound statement where both the implications \( p \rightarrow q \) and \( q \rightarrow p \) must hold truth. For instance, the statement 'The campus is closed if and only if it is Sunday' asserts that both conditions—campus being closed and it being Sunday—are either both true or both false, establishing a bidirectional condition.

Logical Connectives

Logical connectives function as the backbone of symbolic logic, they are symbols or words that connect simple statements to form more complex expressions, known as compound statements. There are several basic logical connectives:

• 'And' (Conjunction), denoted by \( \wedge \)
• 'Or' (Disjunction), denoted by \( \vee \)
• 'Not' (Negation), denoted by \( eg \)
• 'If...then...' (Implication), denoted by \( \rightarrow \)
• 'If and only if' (Biconditional), denoted by \( \leftrightarrow \)

Understanding these connectives is vital to interpret and create compound statements accurately in mathematics.

Negation

The concept of negation is tied to the 'not' logical connective, symbolized by \( eg \) or '~'. Negation is used to invert the truth value of a given statement. For instance, if the original statement \( p \) represents 'The campus is closed', then \( eg p \) (or '~p') indicates that the campus is not closed. Negation plays a critical role in constructing compound statements, especially when defining the conditions under which a statement is false.

Compound Statements

Compound statements combine two or more simple statements using logical connectives to articulate more complex conditions. They can encompass a mix of conjunctions, disjunctions, negations, implications, and biconditionals. In symbolic logic, they are denoted by combining the symbols that represent each part of the statement, resulting in an expressive formula that captures the full meaning of the combined conditions.

In our example, ‘The campus is not closed if and only if it is not Sunday,’ we create a compound statement using negation and biconditional connectives. The symbolic form of this compound statement is \( eg p \leftrightarrow eg q \) which illustrates the direct correlation between the state of the campus being open and the day not being Sunday.