Problem 17 - Thinking Mathematically

Question

Complete the truth table for the given statement by filling in the required columns. $$ \begin{aligned} &\sim p \wedge p\\\ &\begin{array}{|c|c|c|} \hline \boldsymbol{p} & \sim \boldsymbol{p} & \sim \boldsymbol{p} \wedge \boldsymbol{p} \\ \hline \mathrm{T} & & \\ \hline \mathrm{F} & & \\ \hline \end{array} \end{aligned} $$

Step-by-Step Solution

Answer

The completed truth table is: \(p = T\), \(\sim p = F\), \(\sim p \wedge p = F\). \(p = F\), \(\sim p = T\), \(\sim p \wedge p = F\).

Step 1: Determine the values for NOT p

The symbol \(\sim\) is a logical operator that reverses the truth value of the proposition it precedes. If the truth value of \(p\) is True(T), then \(\sim p\) will be False(F). If the truth value of \(p\) is False(F), then \(\sim p\) will be True(T). Therefore, for \(p = T\), \(\sim p = F\) and for \(p = F\), \(\sim p = T\).

Step 2: Determine the values for NOT p AND p

The logical operator '\(\wedge\)' represents the conjunction AND. The resulting statement, A \(\wedge\) B, only holds true if BOTH A and B are true. Hence, the expression \(\sim p \wedge p\) will only be True if both \(\sim p\) and \(p\) are True. But in this case that is impossible since they are always the opposite of each other. Thus \(\sim p \wedge p\) will always be False regardless of the value of \(p\). For \(p = T\), and \(\sim p = F\), \(\sim p \wedge p = F\). For \(p = F\), and \(\sim p = T\), \(\sim p \wedge p = F\).

Step 3: Fill in the truth table

After determining the values for \(\sim p\) and \(\sim p \wedge p\), the cells in the truth table can be filled in. \(p = T\), \(\sim p = F\), \(\sim p \wedge p = F\). \(p = F\), \(\sim p = T\), \(\sim p \wedge p = F\).

Key Concepts

Logical Operators

Logical operators are the backbone of propositional logic, enabling us to construct complex statements from simple ones. The most common logical operators include NOT, represented by the symbol \(\sim\), AND, symbolized by \(\wedge\), and OR, denoted with \(\vee\). Each operator applies a specific rule to the truth values of propositions to produce a new truth value. For example, the NOT operator inverts the truth value: it changes True to False, and False to True. Understanding these operators allows us to analyze and construct truth tables, which are visual representations of all possible outcomes of logical statements.

Logical Conjunction

The logical conjunction, or simply AND, is a vital operation in logic, represented by the symbol \(\wedge\). It relates two propositions and results in a truth value of True only if both propositions are True. In other words, \(A \wedge B\) is True if both \(A\) and \(B\) are True; otherwise, it is False. This operator is crucial for constructing mathematical statements, programming conditions, and logical arguments that depend on the concurrent truth of multiple conditions.

Truth Values

In propositional logic, truth values are the foundation of understanding how logic operates. There are only two possible truth values: True (T) and False (F). A proposition is a statement that can either be True or False but not both. The truth value of a compound statement, constructed with logical operators, is determined by the truth values of its constituent propositions. Truth tables are a systematic way to list all possible combinations of truth values for given propositions and determine the truth values of compound statements based on those combinations.

Propositional Logic

Propositional logic is a branch of logic that deals with statements that can be assigned a truth value. The focus is on how these propositions can be combined using various logical operators to form more complex statements whose truth values depend on the original propositions. The goal of propositional logic is to derive new truths from given propositions and to assess the validity of logical arguments. Truth tables, like the one in our exercise, are an effective tool for analyzing the logic of propositions and ensuring a better understanding of logical deduction.