Problem 27 - Thinking Mathematically

Question

Construct a truth table for the given statement. \((p \leftrightarrow q) \rightarrow \sim r\)

Step-by-Step Solution

Answer

The truth table for the given statement is: \n \[\begin{array}{ccc|c|c|c} p & q & r & p \leftrightarrow q & \sim r & (p \leftrightarrow q) \rightarrow \sim r \\ \hline T & T & T & T & F & F \\ T & T & F & T & T & T \\ T & F & T & F & F & T \\ T & F & F & F & T & T \\ F & T & T & F & F & T \\ F & T & F & F & T & T \\ F & F & T & T & F & F \\ F & F & F & T & T & T \\ \end{array} \]

Step 1 : Prepare the basic inputs (p, q and r)

List all possible values for p, q and r variables. Since there are 3 variables, it will give \(2^3 = 8\) combinations. It starts from (T, T, T) to (F, F, F)

Step 2 : Evaluate the bidirectional condition

A bidirectional condition (p ↔ q) is True when both p and q are either True or False. Write down the results of these 8 situations in the next column of the table.

Step 3 : Evaluate the negation

In the next column, evaluate the negation for r (\(\sim r\)). This will be True if r is False, and False if r is True.

Step 4 : Evaluate the logical implication

Finally put it all together to evaluate \((p \leftrightarrow q) \rightarrow \sim r\). It is only False if \((p \leftrightarrow q)\) is True and \(\sim r\) is False, in all other cases it's True.

Key Concepts

Logical Implication

Understanding logical implication, also known as a conditional statement, is a crucial step in constructing truth tables. In logical implication, the statement has the form of 'if p then q', denoted as \( p \rightarrow q \). The statement \( p \rightarrow q \) is only false when p is true and q is false; in all other combinations, it is true. This might seem counterintuitive because if p happens (is true) but q does not follow (is false), the implication has been violated, hence it is false. If p does not happen, there's no condition to fulfill, and thus, the implication stands.

When constructing a truth table, you will list all possible combinations of truth values for p and q and then determine the truth value of \( p \rightarrow q \) for each combination. This is a critical component in understanding the broader logical expressions that use implication as one part of their structure.

Bidirectional Condition

A bidirectional condition, symbolized by \( p \leftrightarrow q \), indicates a two-way logical relationship. This condition states that both statements p and q will either be both true or both false. It's the logical equivalence of saying, 'p if and only if q.' In a truth table, you would assess the bidirectional condition by checking if p and q have the same truth value for each possible combination.

In the context of the original exercise where you have the expression \( p \leftrightarrow q \) involved, this bidirectional condition is true when p and q are both true (T ↔ T = T) or both false (F ↔ F = T). In any other scenario, the condition would be false. This establishes the foundation for compound statements, leading up to more complex logical structures.

Logical Negation

Logical negation is a simple yet powerful concept in logic. It is the process of inverting the truth value of a statement. If a statement p is true, then the negation of p, symbolized as \( \sim p \), is false, and vice versa. Applying this to truth tables, each time you have a column representing a variable, you should also consider a column for its negation.

For instance, when constructing the truth table for the logical expression involving \( \sim r \), you create a column where every true value (T) for r becomes false (F) in the \( \sim r \) column, and every false value (F) becomes true (T). This fundamental concept enables the understanding of more complex logical constructions by allowing the reversal of truth conditions.

Truth Table Combinations

When confronted with the task of constructing truth tables, one must grapple with various combinations of truth values. A statement with 'n' different variables will have \( 2^n \) combinations of truth values, as each variable can be independently true or false. For the exercise at hand with three variables, p, q, and r, there are \( 2^3 = 8 \) different truth value combinations to consider.

In practice, you systematically list these combinations, often starting with all variables being true and methodically working down to all being false. Once all possible combinations are laid out, the truth values for compound logical expressions can be computed across each row. This not only aids in understanding the relationships between different logical operators but also establishes a clear visual framework from which one can deduce logical outcomes.