QUIZ | TruthTables, Equivalent Statements, and Tautologies

2 TRUTHS AND 1 LIE | DETERMINE THE LIE

• a.) Truth tables help us determine the truth value of compound statements based on the truth values of their components.

• b.) Truth tables can only be used with statements involving three or fewer variables; more complex statements require other methods.

• c.) Each possible combination of truth values for the variables in a compound statement is listed as a separate row in a truth table.

Answer explanation

 Truth tables can handle any number of variables, though the table size grows exponentially with each added variable.

2 TRUTHS AND 1 LIE | DETERMINE THE LIE

• a.) The Column labeled p^q (representing the logical AND of p and q) is true only when both p and q are true.

• b.) The column labeled ~(p^q) represents the negation of p^q and is true in all cases except when both p and q are true.

• c.) The column labeled ~p is true whenever p is true, as it represents the logical complement of p.

Answer explanation

The column labeled ~p is true when ~p is true when p is false, not when p is true, as it represents the negation of p.

2 TRUTHS AND 1 LIE | DETERMINE THE LIE

• a.) Tautologies are always true, regardless of the truth values of their components.

• b.) A contradiction is a statement that is sometimes false. 

• c.) The conjunction of a tautological statement and a contradiction results in a contradiction.

Answer explanation

A contradiction is always false.

2 TRUTHS AND 1 LIE | DETERMINE THE LIE

• a.) If a statement is a tautology, its truth value is true for all possible truth assignments.

• b.) The conjunction of any given statement and a tautological statement is always false.

• c.) A contradiction cannot be true under any circumstance.

Answer explanation

The conjunction of a given statement and a tautological statement is equivalent to the given statement itself.

2 TRUTHS AND 1 LIE | DETERMINE THE LIE

• a.) If two statements have the same truth value for all possible truth assignments to their variables, they are logically equivalent.

•   b.)  De Morgan's law allows us to rewrite negations of conjunctions and disjunctions as equivalent statements involving the opposite logical operation.

• c.)  Logically equivalent statements always look the same when written out in words.

2 TRUTHS AND 1 LIE | DETERMINE THE LIE

• a.)  The inverse of a statement is always logically equivalent to the original statement.

• b.)  The converse of a statement is not necessarily equivalent to the original statement.

•  c.)  The contrapositive of a statement is logically equivalent to the original statement.