91Ó°ÊÓ

Propositional variables are the building blocks of propositional logic. They are symbols, usually represented by letters such as p, q, and r, that stand for specific propositions or statements. Each propositional variable can either be true (T) or false (F). These truth values are fundamental in determining the overall truth of compound propositions.
For example, in the exercise where we work with the variables p, q, and r, each one can independently be true or false. This gives us 8 possible combinations of truth values when considering all three variables together.

• (T, T, T)
• (T, T, F)
• (T, F, T)
• (T, F, F)
• (F, T, T)
• (F, T, F)
• (F, F, T)
• (F, F, F)

Understanding how these variables interact in logical operations is crucial for working through propositional logic problems.

A truth table is a tool used to systematically determine the truth value of a compound proposition for all possible truth values of its constituent propositional variables. It lists all possible combinations of truth values of the variables and the resultant truth value of the compound proposition.
To show that two propositions are logically equivalent, we use a truth table to compare their truth values for every possible combination of truth values of the variables involved. For our exercise, the truth table includes columns for p, q, r, as well as the intermediate and final expressions like \(p \rightarrow r\), \(q \rightarrow r\), \((p \rightarrow r) \vee \ (q \rightarrow r)\), \(p \wedge q\), and \((p \wedge q) \rightarrow r\).
By filling out this table, we can visually confirm if the columns for two expressions match and hence, determine their logical equivalence.

The implication rule in propositional logic is expressed by the operator \( \rightarrow \). A proposition \(p \rightarrow r\) (read as 'p implies r') is false only if p is true and r is false. In all other cases, the implication is true.
For example, consider \(p = T\) and \(r = F\). Here, \(p \rightarrow r = F\) because we have a true antecedent leading to a false consequent. However, if p were false regardless of r's truth value, or if both were true, the implication would be true.
This rule is essential for computing truth values in implication expressions in our truth table and plays a central role in determining the logical equivalence of propositions.

Disjunction is another fundamental logical operation, denoted by the operator \( \vee \). It represents an 'OR' operation. A disjunction \(p \vee q\) is true if at least one of p or q is true. It is false only when both p and q are false.
For the expression \((p \rightarrow r) \vee (q \rightarrow r)\), we need to find the truth value of \(p \rightarrow r\) and \(q \rightarrow r\) individually and then apply the disjunction rule. If either \(p \rightarrow r\) or \(q \rightarrow r\) is true, the result will be true.
Understanding disjunction helps us correctly fill out the truth table and determine the truth value of compound propositions involving 'OR'.

Conjunction is represented by the operator \( \wedge \) and signifies an 'AND' operation. A conjunction \(p \wedge q\) is true only if both p and q are true. It is false if either or both of the variables are false.
In our exercise, we calculate the truth value of \(p \wedge q\) to use it in further compound propositions such as \((p \wedge q) \rightarrow r\). The conjunction provides a basis for how these variables combine together under the 'AND' operation.
This is crucial because recognizing these truth values helps us construct and compare the necessary columns in the truth table to show logical equivalence.