A truth table is a mathematical table used in logic, e.g. propositions. A truth table can be used to determine whether a proposition is true or false
There are several basic truth tables. In this module, we discuss three truth tables.
- Negation of a Proposition
Let p be a proposition, the negation of p, denoted as ~p (or ¬p), is the statement “not p”. The truth value of the negation of p, ~p, is the opposite of the truth value of p.
For example, let the proposition p be “Today is Friday”, the negation of p, ~p, is “Today is NOT Friday”. On the other hand, let the proposition p be “Today is NOT Friday,” the negation of p, ~p, is “Today is Friday.”
Another example, let the proposition p be “It is raining outside”, the negation of p, ~p, is “It is NOT raining today”. On the other hand, let the proposition p be “It is NOT raining today,” the negation of p, ~p, is “It is raining today”
The following table is the truth table for the negation of a proposition p. This table has a row for each of the two possible truth values of a proposition p. Each row shows the truth value of ~p corresponding to the truth value of p for this row.
| Table 1: The Truth Table for the Negation of a Proposition | |
| p | ~p |
| T | F |
| F | T |
- Conjunction of Propositions
Let p and q be propositions. The conjunction of p and q, denoted as p^q, is the proposition “p and q.” The conjunction p ^ q is true when both p and q are true and is false otherwise.
For example, let p be “Today is Friday” and q be “It is raining today,” then the conjunction p ^ q is “Today is Friday and It is raining today.” This proposition is true on rainy Friday and is false on any day that is not a Friday and on Fridays when it does not rain.
Another example, let p be “Today is NOT Friday” and q be “It is NOT raining today”, then the conjunction p^q is “Today is NOT Friday and It is NOT raining today”. This proposition is true on any day other than rainy Friday, and is false on rainy Friday.
The following table is the truth table for the conjunction of proposition p and q.
| Table 2: The Truth Table for the Conjunction of Two Propositions | ||
| p | q | p ^ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
- Disjunction of Propositions
Let p and q be propositions. The disjunction of p and q, denoted as p?q, is the proposition “p or q”. The disjunction p?q is false when both p and q are false and is true otherwise.
For example, let p be “Today is Friday” and q be “It is raining today,” then the disjunction p?q is “Today is Friday or It is raining today.” This proposition is true on Fridays or any rainy days and is false on any non-rainy day at is not a Friday.
The following table is the truth table for the disjunction of proposition p and q.
| Table 3: The Truth Table for the Disjunction of Two Propositions | ||
| p | q | p?q |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
To learn more about truth tables, check the following sites:
- Truth tables demo introduction http://www.youtube.com/watch?v=98jBy8x6HHw
- Truth table for basic logic http://www.youtube.com/watch?v=SRzSZ_rEE_A
To learn more about tautology and contradiction, check the following sites:
- Tautology (logic) http://en.wikipedia.org/wiki/Tautology_%28logic%29
- Tautology and Contradiction http://people.umass.edu/partee/409/Deduction_I.pdf
