84 lines
2.6 KiB
Markdown
84 lines
2.6 KiB
Markdown
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---
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title: Logical Operators
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---
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## Logical Operators
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**AND (&&)** </br>
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A&&B returns True if both A and B are True. If either A or B (or both) are False, then A&&B is False.
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| A | B |AND(A,B)|
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|---|---|---|
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| f| t | f |
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| f| f | f |
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| t| t | t |
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| t| f | t |
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**OR (||)** </br>
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A||B returns True if A or B (or both A and B) is True. Only returns False if both A and B is False.
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| A | B |OR(A,B)|
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|---|---|---|
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| f| t | t |
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| f| f | f |
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| t| t | t |
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| t| f | t |
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**NOT (!)** </br>
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Returns the opposite value. Ex. if A is true, then !A is false, and if A is false, then !A is true. This is the only logical operator that works on only one input, which makes it a unary operator.
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| A | B |NOT(A)| NOT(B)
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|---|---|---|---|
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| f| t | t |f|
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| f| f | t |t|
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| t| t | f |f|
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| t| f | f |t|
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**XOR ("eXclusive or")** </br>
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Is known as **exclusive or**. Similar to OR, but returns False if both A and B are true. That is, XOR returns true if one and only one of A or B is True.
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| A | B |XOR(A,B)|
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|---|---|---|
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| f| t | t |
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| f| f | f |
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| t| t | f |
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| t| f | t |
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**Implication (A -> B)** </br>
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Read as "if A, then B" or "A implies B".
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Only returns False when A is True and B is False. Otherwise the implication is True. </br>
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<img src="http://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables13.png">
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Note: Implications are often used for direct mathematical proofs. A represents the hypothesis, while B is the conclusion.
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The only time the conditional is false is when a true value leads to a false value.
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| A | B |IF(A,B)|
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|---|---|---|
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| f| t | t |
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| f| f | t |
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| t| t | t |
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| t| f | f |
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**Logical Equivalence (iff: if and only if)** </br>
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"P if and only if Q" is the same as "P implies Q AND Q implies P". In other words, the truth tables for P and Q are identical for all truth values.
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This is known as the biconditional. It is the equivalent of A -> B **AND** B->A. It means that both conditionals must be satisfied in order for the biconditional to be true.
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You can easily see that the out output of the IFF operator in the truth table is the same as the ANDing of columns 3 and 4.
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| A | B |IF(A,B)| IF(B,A)| IFF(A,B)|
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|---|---|---|---|---|
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| f| t | t |f|f|
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| f| f | t |t|t|
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| t| t | t |t|t|
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| t| f | f |t|f|
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#### More Information:
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<!-- Please add any articles you think might be helpful to read before writing the article -->
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+* [Logical Operators in Javascript](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Operators/Logical_Operators)
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+* [Logical Operators in PHP](http://php.net/manual/en/language.operators.logical.php)
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+* [Logical Operators in C++](http://en.cppreference.com/w/cpp/language/operator_logical)
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