Problem 19 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

After filling in the truth table, the rows are completed as follows: for \(p=T, q=T\), \(\sim p = F\) and \(\sim p \land q = F\); for \(p=T, q=F\), \(\sim p = F\) and \(\sim p \land q = F\); for \(p=F, q=T\), \(\sim p = T\) and \(\sim p \land q = T\); for \(p=F, q=F\), \(\sim p = T\) and \(\sim p \land q = F\).

Step 1: Derive the value of \(\sim p\)

Since \(\sim p\) is the negation of \(p\), whenever \(p\) is True, \(\sim p\) is False, and vice versa. Thus you can fill the column \(\sim p\) as follows: when \(p = T\), then \(\sim p = F\); when \(p = F\), then \(\sim p = T\). This will lead to the following table: \[ \begin{{array}}{{cc|c|c}}p & q & \sim p & \sim p \land q\\hlineT & T & F & \T & F & F & \F & T & T & \F & F & T & \\end{{array}} \]

Step 2: Derive the value of \(\sim p \land q\)

The \(\sim p \land q\) is True only if both \(\sim p\) and \(q\) are True, else it is False. Apply this rule to the truth table to fill the last column: when \(\sim p = F, q = T\), then \(\sim p \land q = F\); when \(\sim p = F, q = F\), then \(\sim p \land q = F\); when \(\sim p = T, q = T\), then \(\sim p \land q = T\); when \(\sim p = T, q = F\), then \(\sim p \land q = F\). Now, the completion of the truth table will be: \[ \begin{{array}}{{cc|c|c}}p & q & \sim p & \sim p \land q\\hlineT & T & F & F\T & F & F & F\F & T & T & T\F & F & T & F\\end{{array}} \]

Key Concepts

Logical Conjunction

In the world of logic, the term 'logical conjunction' refers to a compound statement formed with the word 'and'. In mathematical logic, it is typically represented by the symbol \( \land \). A logical conjunction between two statements is true only if both statements are independently true. For instance, if we consider two propositions, \( p \) and \( q \) , the conjunction \( p \land q \) requires both \( p \) and \( q \) to be true for the entire expression to be considered true.

When we lay out the logic of this operation in a truth table, we look at all the possible truth values of \( p \) and \( q \) and note down whether the conjunction holds. This is an essential operation in Boolean algebra and plays a crucial role in constructing and understanding logical arguments and computational logic circuits.

Logical Negation

Logical negation is a fundamental concept in logic that reflects the opposite or denial of a statement. When we negate a statement—symbolized as \( \sim \) or sometimes as \( eg \)—we are creating a new proposition that will be true when the original statement is false, and false when the original statement is true.

For example, if we have a proposition \( p \), and we apply logical negation to get \( \sim p \) (read as 'not \( p \)'), we are effectively stating that whatever truth value \( p \) holds, \( \sim p \) will hold the opposite. This operation of flipping the truth value is the cornerstone of constructing more complex logical expressions and is widely used in mathematical proofs, computer programming, and digital circuit design.

Logic in Mathematics

Logic is an indispensable part of mathematics, serving as the foundation on which mathematical statements and proofs are built. It provides a systematic way to deduce new information from known facts and to assess the validity of arguments. Mathematical logic uses symbols and formal methods to depict logical expressions unambiguously, allowing for clear communication of complex ideas.

Moreover, logic in mathematics extends beyond just symbols; it includes the study of formal systems, proof theory, and model theory. Within these systems, the operation of mathematical facts can be distilled into precise statements, and their interactions can be rigorously analyzed. Students and scholars alike delve into logic to gain a better understanding of mathematical reasoning and computational theory, making it one of the most vital and far-reaching fields of study.

Boolean Algebra

Boolean algebra is a branch of algebra that deals with variables that have two possible values: true or false. This simplification into a binary system allows for the manipulation and combination of truth values through logical operations, such as conjunction (AND), disjunction (OR), and negation (NOT).

In the context of a truth table, Boolean algebra uses algebraic methods to represent and simplify logical expressions. The truth table serves as a tool to visualize the outcomes of Boolean operations for all possible input combinations. Consequently, Boolean algebra is not only central to logic and digital circuit design but is also widely employed in computer science, set theory, and statistics to analyze and optimize various systems.