129 lines
3.5 KiB
Markdown
129 lines
3.5 KiB
Markdown
---
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title: Truth Tables
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---
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## Truth Tables
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A truth table is a mathematical tool used in Boolean Algebra. It consists of a column each for the function variables. A final column holds the functional value evaluated for the corresponding values of the variables. For a boolean function of n variables, its truth table expansion will have 2^n rows. This is beacuse each variable has two possible states – true & false.
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### AND
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Let us explore the truth table for the AND operator:
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| x | y | x AND y |
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|---|---|---|
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| F | F | F |
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| F | T | F |
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| T | F | F |
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| T | T | T |
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AND is binary operator. It operates on two variables, say `x`, `y`.
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Thus we have 2^2 = 4 columns in our truth table!
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The last column is the functional value – x AND y.The logic for AND operation is that if values of x and y are both True only then the output would have the value True else it would be False.
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Similarly truth tables for other logical operators -
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### NOT
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| x | NOT X |
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|---|---|
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| F | T |
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| T | F |
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### OR
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| x | y | x OR y |
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|---|---|---|
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| F | F | F |
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| F | T | T |
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| T | F | T |
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| T | T | T |
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### XOR
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| x | y | x XOR y |
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|---|---|---|
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| F | F | F |
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| F | T | T |
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| T | F | T |
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| T | T | F |
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OR operator:
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| x | y | x OR y |
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|---|---|---|
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| F | F | F |
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| F | T | T |
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| T | F | T |
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| T | T | T |
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NOT operator:
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| x | NOT x |
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|---|---|
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| F | T |
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| T | F |
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Implication operator:
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| x | y | x IMPLY y |
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|---|---|---|
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| F | F | T |
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| F | T | T |
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| T | F | F |
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| T | T | T |
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The implication operator can often be confusing to some. It is useful to relate real world examples to aid understanding for this operator. For example, consider:
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If it is raining then I use an umbrella.
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Here, assuming it is raining, then I use an umbrella (statement holds)
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But if it is raining, and I don't use an umbrella, then the statement fails to hold.
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Despite that, if it is not raining, and I still use an umbrella, then the statement also holds (it doesn't really matter if the umbrella is used or not, since it's not raining. Although it would look rather strange).
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However, the implication operator can be puzzling for propositions involved that are false in the real world. Consider:
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If the sun is made out of water then 1 + 1 = 3.
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According to the implication truth table this propositional formula is true.
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P implies Q can also be thought as an abbreviation for NOT(P) OR Q.
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Double implication operator:
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| x | y | x <-> y |
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|---|---|---|
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| F | F | T |
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| F | T | F |
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| T | F | F |
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| T | T | T |
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Truth tables are a powerful tool. They can be used to express and evaluate simple boolean functions and operations, complex combinational circuits, and sequential logic circuits!
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Here is the truth table for the OR operator
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| x | y | x OR y |
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|---|---|---|
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| F | F | F |
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| F | T | T |
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| T | F | T |
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| T | T | F |
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Just like above the OR operator operates on two variables, notice that the only time the OR operator evaluates to True is when `x` & `y` negate eachother.
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Let's do one more, let's do the table for the Negation, this operates on one value instead of two
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| x | NOT x |
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|---|---|
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| T | F |
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| F | T |
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This rule is simpler and it simply negates the original value of `x`
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#### More Information:
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- <a href='http://hyperphysics.phy-astr.gsu.edu/hbase/Electronic/truth.html' target='_blank' rel='nofollow'>Hyperphysics - Georgia State University</a>
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- <a href='https://en.wikipedia.org/wiki/Truth_table' target='_blank' rel='nofollow'>Wikipedia</a>
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