Example: Boolean Operators

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Evaluating Boolean Expression

Evaluate Boolean expressions by inserting a Boolean expression followed by the evaluation operator.

The Boolean equal to operator is different from the evaluation operator. When evaluating a Boolean expression, 1 denotes "true" and 0 denotes "false."

Logical Operators

Define two vectors.

Use the vectorize operator to evaluate the following Boolean expressions term by term:

NOT—Negation	<region id="ID0EAV4Q" top="504" left="120" width="70.0416666666667" height="64.331" xmlns="http://schemas.mathsoft.com/worksheet50"> <math> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <vectorize /> <parens> <apply> <not /> <id>P</id> </apply> </parens> </apply> <result> <matrix rows="4" cols="1"> <real>0</real> <real>0</real> <real>1</real> <real>1</real> </matrix> </result> </eval> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" /> </resultFormat> </math> </region>
AND—Conjunction	<region id="ID0E6V4Q" top="576" left="120" width="80.4583333333333" height="64.331" xmlns="http://schemas.mathsoft.com/worksheet50"> <math> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <vectorize /> <parens> <apply> <and /> <id>P</id> <id>Q</id> </apply> </parens> </apply> <result> <matrix rows="4" cols="1"> <real>1</real> <real>0</real> <real>0</real> <real>0</real> </matrix> </result> </eval> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" /> </resultFormat> </math> </region>
OR—Disjunction	<region id="ID0E5W4Q" top="648" left="120" width="80.4583333333333" height="64.331" xmlns="http://schemas.mathsoft.com/worksheet50"> <math> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <vectorize /> <parens> <apply> <or /> <id>P</id> <id>Q</id> </apply> </parens> </apply> <result> <matrix rows="4" cols="1"> <real>1</real> <real>1</real> <real>1</real> <real>0</real> </matrix> </result> </eval> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" /> </resultFormat> </math> </region>
XOR—Exclusive or	<region id="ID0E2X4Q" top="720" left="120" width="82.6583333333333" height="64.331" xmlns="http://schemas.mathsoft.com/worksheet50"> <math> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <vectorize /> <parens> <apply> <xor /> <id>P</id> <id>Q</id> </apply> </parens> </apply> <result> <matrix rows="4" cols="1"> <real>0</real> <real>1</real> <real>1</real> <real>0</real> </matrix> </result> </eval> <resultFormat> <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" /> <matrix size="12,12" offset="0,0" /> </resultFormat> </math> </region>

Logical Equivalence

You can show that P ⋁ (Q ∧ R) and (P ⋁ Q) ∧ (P ⋁ R) are logically equivalent.