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In logic, propositional variables are symbols that represent specific propositions, which can be either true or false. Commonly, these variables are denoted by letters such as \(p\), \(q\), and \(r\).
Each propositional variable can take on one of two truth values:

• True (T)
• False (F)

Propositional variables are the basic building blocks in constructing logical expressions and compound propositions. For example, if \(p\) represents 'It is raining,' \(p\) can be either true (if it is indeed raining) or false (if it is not raining).
Understanding propositional variables is crucial for building more complex logical statements and generating truth tables.

Compound propositions are formed by combining two or more propositional variables using logical connectives such as AND, OR, NOT, IF-THEN, and IF AND ONLY IF.
Here are some examples:

• \textbf{Conjunction (AND):} \text{If \(p\) is 'It is raining' and \(q\) is 'I have an umbrella,' then the compound proposition \(p \text{ AND } q\) is true if both statements are true.}
  
• \textbf{Disjunction (OR):} \text{If \(p\) is 'It is raining' and \(q\) is 'I have an umbrella,' then the compound proposition \(p \text{ OR } q\) is true if at least one of the statements is true.}
  
• \textbf{Negation (NOT):} \text{If \(p\) is 'It is raining,' then the compound proposition \(\text{NOT } p\) is true if \(p\) is false (i.e., it is not raining).} Compound propositions allow us to explore more complex logical relationships and are crucial in generating multiple scenarios within a truth table.

Truth values indicate whether a given proposition is true or false. In binary logic, every proposition must have one and only one truth value:

• True (T)
• False (F)

Truth tables use these values to systematically explore all possible scenarios for a set of propositions.
For example, if we have two propositional variables, \(p\) and \(q\), each can be either true or false. As a result, we have four possible combinations of truth values for \(p\) and \(q\): (T, T), (T, F), (F, T), and (F, F).

In the context of a truth table for compound propositions, the truth values help determine the outcome of the entire expression based on the truth values of its component variables. By analyzing these scenarios, you can understand how different logical operators interact and affect the overall truth value of a compound proposition.