Difference between revisions of "Truth table proofs"
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| − | Truth tables are used to show all possible values that a given logical expression might take. | + | Truth tables are used to show all possible values that a given logical expression might take. |
| + | |||
| + | =Examples= | ||
| + | ==Definition of AND== | ||
| + | The following gives the definition of the logical AND operation. | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 14: | Line 18: | ||
|} | |} | ||
| − | A truth table can also be used to prove a logical identity. The following proves | + | ==Proof of one of De Morgan's laws== |
| + | A truth table can also be used to prove a logical identity. The following proves De Morgan's law that ¬ (A ∧ B) is equivalent to (¬ A) ∨ (¬ B). Notice that the truth values for these (the last two columns in the table) are always the same. | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
| − | ! A !! B !! A ∧ B !! ¬ (A ∧ B) !! (¬ A) ∨ (¬ B) | + | ! A !! B !! A ∧ B !! ¬ (A ∧ B) !! (¬ A) ∨ (¬ B) |
|- | |- | ||
| false || false || false || true || true | | false || false || false || true || true | ||
| Line 28: | Line 33: | ||
| true || true || true || false || false | | true || true || true || false || false | ||
|} | |} | ||
| + | |||
| + | ==Disproof of a logical fallacy== | ||
| + | Truth tables can also be used to disprove a logical fallacy. For example, one fallacy is to assume the converse of an implication holds. If you know that (A → B) is true, and you know that B is true, the fallacy would be to then conclude that A must also be true. Note that the statement - "if an animal is a mammal then it is also a vertebrate" - is true. If we have an animal that is a vertebrate (for example, a dog), it would be a fallacy to now conclude that the animal must also be a mammal. | ||
| + | |||
| + | Consider this truth table. | ||
| + | {| class="wikitable" | ||
| + | |- | ||
| + | ! A !! B !! A → B (which is equivalent to ¬ A ∨ B) | ||
| + | |- | ||
| + | | false || false || true | ||
| + | |- | ||
| + | | false || true || true | ||
| + | |- | ||
| + | | true || false || false | ||
| + | |- | ||
| + | | true || true || true | ||
| + | |} | ||
| + | In the | ||
| + | |||
| + | |||
| + | =Assignment= | ||
Revision as of 15:03, 29 August 2022
Truth tables are used to show all possible values that a given logical expression might take.
Contents
Examples
Definition of AND
The following gives the definition of the logical AND operation.
| A | B | A ∧ B |
|---|---|---|
| false | false | false |
| false | true | false |
| true | false | false |
| true | true | true |
Proof of one of De Morgan's laws
A truth table can also be used to prove a logical identity. The following proves De Morgan's law that ¬ (A ∧ B) is equivalent to (¬ A) ∨ (¬ B). Notice that the truth values for these (the last two columns in the table) are always the same.
| A | B | A ∧ B | ¬ (A ∧ B) | (¬ A) ∨ (¬ B) |
|---|---|---|---|---|
| false | false | false | true | true |
| false | true | false | true | true |
| true | false | false | true | true |
| true | true | true | false | false |
Disproof of a logical fallacy
Truth tables can also be used to disprove a logical fallacy. For example, one fallacy is to assume the converse of an implication holds. If you know that (A → B) is true, and you know that B is true, the fallacy would be to then conclude that A must also be true. Note that the statement - "if an animal is a mammal then it is also a vertebrate" - is true. If we have an animal that is a vertebrate (for example, a dog), it would be a fallacy to now conclude that the animal must also be a mammal.
Consider this truth table.
| A | B | A → B (which is equivalent to ¬ A ∨ B) |
|---|---|---|
| false | false | true |
| false | true | true |
| true | false | false |
| true | true | true |
In the
