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A truth table is a systematic way to illustrate and analyze how different truth values for certain propositions determine the truth value of a compound statement made using logical operators. It plays a crucial role in evaluating the validity of logical expressions in propositional logic.

When constructing a truth table, one begins by listing all possible combinations of truth values for the given propositional variables. For example, if you have two variables, such as in our exercise featuring variables 'p' and 'q', you would have four combinations: TT, TF, FT, and FF. These represent all the possible states the variables can have.

Next, one column for each sub-expression is calculated based upon the truth values and the nature of the logical operator within that sub-expression. Finally, the truth value of the entire expression is deduced. This makes it easy to see in which scenarios the compound statement is true or false. In the case of validating a tautology, a statement is considered a tautology if, and only if, the final column (representing the compound statement) consists solely of true values.

Logical operators, also known as logical connectives, are the symbols that connect propositions or statements to form more complex logical expressions. In propositional logic, the basic logical operators include 'and' (conjunction), 'or' (disjunction), 'not' (negation), 'if-then' (implication), and 'if and only if' (biconditional).

Conjunction \(\wedge\)

The operator 'and' pairs two propositions to be true only if both are true. For example, 'p \wedge q' is true only if both 'p' and 'q' are true.

Implication \(\rightarrow\)

The 'if-then' operator is a bit more complex; 'p \rightarrow q' translates to 'if p then q'. It is only false when the first proposition 'p' is true, and the second proposition 'q' is false. Otherwise, it is considered true.

In our exercise, these logical operators are used to develop a more intricate statement that can be analyzed with the help of a truth table to check for tautology.

Propositional variables are the basic elements of propositional logic. These are symbols that represent propositions, which are statements that can be either true or false. In the context of our exercise, 'p' and 'q' are propositional variables.

Each propositional variable can take on a truth value of 'true' (T) or 'false' (F). When we combine them using logical operators, we can create compound logical expressions. These expressions' truth values depend on the truth values assigned to each variable and how those variables are connected through logical operators.

Understanding each variable and the role it plays in the overall expression is essential when constructing truth tables and determining whether an expression forms a tautology.