Truth Tables: a AND b

Date: 09/10/2001 at 22:43:08
From: Emilee Longfellow
Subject: Truth Tables

I am having trouble finding out what truth tables are.

Thank you for your help,
Emilee

Date: 09/11/2001 at 03:36:36
From: Doctor Jeremiah
Subject: Re: Truth Tables

Hi Emilee:

Truth tables are a way of showing the answer to a question by listing 
every possible value of every input and giving each the correct 
answer.

For example let's take the operation called "and."

When you say "when a is true AND b is true ..."
what do you mean?

What do you mean when b is not true and a is true?
What do you mean when neither one is true?

To show what the answer is for every possible value
of a and b we need to make a truth table:

   a   |   b   |  a AND b
 ------+-------+----------
 false | false |
 false | true  |
 true  | false |
 true  | true  |

See how we made a row for each possible combination of a and b?  Now 
we just need the answers (for the third column)

If a is false and b is false then what is a AND b? Is it true or 
false? For example, a could be the statement "grass is purple" (which 
is obviously false) and b could be the statement "sky is green" (which 
is obviously false)  So when we say "grass is purple AND  sky is 
green" would it be false or true?  False, right?

   a   |   b   |  a AND b
 ------+-------+----------
 false | false |   false
 false | true  |
 true  | false |
 true  | true  |

If a is false and b is true then what is a AND b? Lets say that a is 
the statement "grass is purple" (which is obviously false) and b is 
the statement "sky is blue" (which is true)  So when we say "grass is 
purple AND sky is blue" would the whole thing be false or true?  
False, right?

   a   |   b   |  a AND b
 ------+-------+----------
 false | false |   false
 false | true  |   false
 true  | false |
 true  | true  |

We keep doing this until we have all the answers filled in:

   a   |   b   |  a AND b
 ------+-------+----------
 false | false |   false
 false | true  |   false
 true  | false |   false
 true  | true  |   true

That's the truth table for "a and b," and whoever looks at it will 
instantly know that when we say two things with an AND in between we 
mean that the whole sentence is true only when both parts are true 
(because every other case is false)

So that's how a truth table works. Let me know if you want to talk 
about this some more.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/