# Using the truth table, examine whether the following statement pattern is tautology, contradiction and contingency.

$(p\wedge \sim q)\leftrightarrow (p\to q)$

Last updated date: 27th Mar 2023

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Hint: Construct the truth table for $(p\wedge \sim q)\leftrightarrow (p\to q)$ and if all the propositions are true in every row then it is tautology, if all the propositions are false in every row then contradiction and if there is at least one row with true condition and one row with false condition, then it is contingency.

Complete step-by-step answer:

A truth table is a mathematical table used in logic- specifically in connection with Boolean algebra, Boolean functions and propositional calculus- which sets out the functional values of logical expressions on each of their functional arguments, that is for each combination of values taken by their logical variables. A truth table can be used to show whether a propositional expression is true for all input values, that is logically valid.

Tautology: A tautology has a logical form that cannot possibly be false no matter what truth values are assigned to the sentence letters.

Contradiction: A tautology has a logical form that cannot possibly be true no matter what truth values are assigned to the sentence letters.

Contingency: A Contingency has a logical form that can be either true or false depending on what truth values are assigned to the sentence letters.

Note: $p\wedge q$ is true when both $p$ and $q$ are true. $p\to q$ is true when $q$ is true or both are false. $p\leftrightarrow q$ is true when both $p$ and $q$ are true or both are false. This is the rule used to draw the above truth table.

Complete step-by-step answer:

A truth table is a mathematical table used in logic- specifically in connection with Boolean algebra, Boolean functions and propositional calculus- which sets out the functional values of logical expressions on each of their functional arguments, that is for each combination of values taken by their logical variables. A truth table can be used to show whether a propositional expression is true for all input values, that is logically valid.

Tautology: A tautology has a logical form that cannot possibly be false no matter what truth values are assigned to the sentence letters.

Contradiction: A tautology has a logical form that cannot possibly be true no matter what truth values are assigned to the sentence letters.

Contingency: A Contingency has a logical form that can be either true or false depending on what truth values are assigned to the sentence letters.

Note: $p\wedge q$ is true when both $p$ and $q$ are true. $p\to q$ is true when $q$ is true or both are false. $p\leftrightarrow q$ is true when both $p$ and $q$ are true or both are false. This is the rule used to draw the above truth table.

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