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Truth Tables:
A STATEMENT - Any statement that can be true (T) or false (F), but never both. For simplification we will denote our statements by lowercase letter variables. For example p, q, etc.
Example of a statement: "Sugar is sweet." Lets denote this true statement by letter p. Another example of a statement: "Water is a fluid." And lets denote this statement by letter q.
Negation of p (not p) is true when p is false and vice versa.
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Negation of a statement (not) |
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p |
not p |
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T |
Sugar is sweet. |
F |
Sugar is not sweet. |
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F |
Sugar is not sweet. |
T |
Sugar is sweet. |
Conjunction (p and q) is true if and only if both of the statements are true.
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Conjunction of two statements (p and q) |
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p |
q |
p and q |
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| T | Sugar is sweet. | T | Water is a fluid. | T | Sugar is sweet and water is a fluid. |
| T | Sugar is sweet. | F | Water is not a fluid. | F | Sugar is sweet and water is not a fluid. |
| F | Sugar is not sweet. | T | Water is a fluid. | F | Sugar is not sweet and water is a fluid. |
| F | Sugar is not sweet. | F | Water is not a fluid. | F | Sugar is not sweet and water is not a fluid. |
Disjunction of p, q (p or q) is true if at least one of the statements is true.
| Disjunction of two statements (p or q) | |||||
| p | q | p or q | |||
| T | Sugar is sweet. | T | Water is a fluid. | T | Sugar is sweet or water is a fluid. |
| T | Sugar is sweet. | F | Water is not a fluid. | T | Sugar is sweet or water is not a fluid. |
| F | Sugar is not sweet. | T | Water is a fluid. | T | Sugar is not sweet or water is a fluid. |
| F | Sugar is not sweet. | F | Water is not a fluid. | F | Sugar is not sweet or water is not a fluid. |
Exclusive or of p, q (p exclusive or q) is true if only one of the two statements is true
| Exclusive or of two statements (p exclusive or q) | |||||
| p | q | p exclusive or q | |||
| T | Sugar is sweet. | T | Water is a fluid. | F | Sugar is sweet or water is a fluid. |
| T | Sugar is sweet. | F | Water is not a fluid. | T | Sugar is sweet or water is not a fluid. |
| F | Sugar is not sweet. | T | Water is a fluid. | T | Sugar is not sweet or water is a fluid. |
| F | Sugar is not sweet. | F | Water is not a fluid. | F | Sugar is not sweet or water is not a fluid. |
Implication of p, q (if p then q) is false only when p is true and q is false.
| Implication of two statements (If p then q) | |||||
| p | q | If p then q | |||
| T | Sugar is sweet. | T | Water is a fluid. | T | If sugar is sweet, then water is a fluid. |
| T | Sugar is sweet. | F | Water is not a fluid. | F | If sugar is sweet, then water is not a fluid. |
| F | Sugar is not sweet. | T | Water is a fluid. | T | If sugar is not sweet, then water is a fluid. |
| F | Sugar is not sweet. | F | Water is not a fluid. | T | If sugar is not sweet, then water is not a fluid. |
Biconditional of p, q (p if and only if q) is true if and only if both of the statements are either true or false. In another words both of the statements must be of the same truth value.
NOTE: Abbreviation for "if and only if" is well known in mathematics as IFF.
| Biconditional of two statements (p iff q) | |||||
| p | q | p iff q | |||
| T | Sugar is sweet. | T | Water is a fluid. | T | Sugar is sweet if and only if water is a fluid. |
| T | Sugar is sweet. | F | Water is not a fluid. | F | Sugar is sweet if and only if water is not a fluid. |
| F | Sugar is not sweet. | T | Water is a fluid. | F | Sugar is not sweet if and only if water is a fluid. |
| F | Sugar is not sweet. | F | Water is not a fluid. | T | Sugar is not sweet if and only if water is not a fluid. |