Problem 45 - Thinking Mathematically

Question

a. Write each statement in symbolic form. Assign letters to simple statements that are not negated. b. Construct a truth table for the symbolic statement in part (a). c. Use the truth table to indicate one set of conditions that makes the compound statement true, or state that no such conditions exist. It is not true that I bought a meal ticket and did not use it.

Step-by-Step Solution

Answer

The symbolic form of 'It is not true that I bought a meal ticket and did not use it' is \(\neg (M \land \neg U)\). This statement is true if: 'I bought a meal ticket and used it', or 'I did not buy a meal ticket but I used it', or 'I did not buy a meal ticket and did not use it'.

Step 1: Symbolic Transformation

Let's denote: \(M\) as 'I bought a meal ticket' and \(U\) as 'I used it'. Rewording the given statement, it becomes 'It is not true that both \(M\) and not \(U\) are true'. In symbolic form, this can be written as: \(\neg (M \land \neg U)\).

Step 2: Constructing the Truth Table

The truth table for two variables \(M\) and \(U\) and the compound statement \(\neg (M \land \neg U)\) can be constructed like this: | \(M\) | \(U\) | \(\neg U\) | \(M \land \neg U\) | \(\neg (M \land \neg U)\) | | T | T | F | F | T | | T | F | T | T | F | | F | T | F | F | T | | F | F | T | F | T | Here, T represents True and F represents False.

Step 3: Analysis of the Truth Table

From the truth table, the compound statement, \(\neg (M \land \neg U)\), is true if: a) \(M\) is true and \(U\) is true b) \(M\) is false and \(U\) is true c) \(M\) is false and \(U\) is false Therefore the conditions that make the statement 'It is not true that I bought a meal ticket and did not use it' true are: a) I bought a meal ticket and used it. b) I did not buy a meal ticket but I used it. c) I did not buy a meal ticket and did not use it.

Key Concepts

Truth Table

A truth table is a valuable tool used in symbolic logic to evaluate all possible truth values of a logical expression or compound statement. By evaluating every possible combination of truth values for the variables involved, we can determine when a statement is true or false.

To construct a truth table, we list all possible truth values for each variable in our expression. For each combination, we determine the truth value of the entire expression.

In our case, we analyzed the expression \(eg (M \land eg U)\), where \(M\) stands for 'I bought a meal ticket', and \(U\) stands for 'I used it'. We examined each combination of truth values for \(M\) and \(U\), calculated \(eg U\), calculated \(M \land eg U\), and finally, the negation of the conjunction, \(eg (M \land eg U)\). This process helps to visualize under what conditions the compound statement takes on the true or false value.

Compound Statement

A compound statement in logic is formed by combining one or more simple statements using logical connectives. Common connectives include 'and' (\(\land\)), 'or' (\(\lor\)), and 'not' (\(eg\)). By combining simple thoughts logically, we form more complex statements which can be analyzed for their truth values.

In our exercise, the compound statement "It is not true that I bought a meal ticket and did not use it" involves the use of logical connectives. Translating this into the symbolic logic form \(eg (M \land eg U)\), we see how negation and conjunction are employed to express this idea. The compound statement thus becomes an anchor for the truth table analysis, opening up an examination of when our statement holds true.

Logical Negation

Logical negation is a fundamental operation in logic which reverses the truth value of a statement. Using the negation symbol \(eg\), a true statement becomes false, and a false statement becomes true.

In our problem, the phrase "It is not true that" triggers the need for negation. If we have a statement like "\(M \land eg U\)", negating it results in \(eg (M \land eg U)\), which expresses the reversed truth conditions.

This operation is crucial for expressing the full meaning of statements in their negative forms within logic. It helps convey the opposite of a given scenario, showing that if one part of the statement isn't met the whole changes its truth value.

Symbolic Representation

Symbolic representation is the use of symbols or letters to denote simple or complex statements. In logic, this gives us a compact and precise way of formulating statements that are potentially long or cumbersome when expressed verbally.

In this exercise, we assigned the letter \(M\) to the statement 'I bought a meal ticket' and \(U\) to 'I used it'. Through these symbols, the original statement "It is not true that I bought a meal ticket and did not use it" was transformed into the symbolic form \(eg (M \land eg U)\).

This transformation aids in logical analysis and manipulation, enabling us to apply logical rules and evaluate statements within a structured framework like a truth table. Symbolic logic thus simplifies the study and application of logical reasoning in problems similar to our example.