# Can anyone explain Truth Tables?

Asked by

IBERnineD (

7269)
February 15th, 2009

I’m having difficulty learning how to create a truth table in math and also understand them! So any tips or help would be great!

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## 4 Answers

I’m not sure how complicated the tables you are dealing with are, so I’ll try to explain the basics, and you can feel free to ask me if you are looking for more.

A logical statement is just that, a statement. Whether this statement is true depends on the facts, in the same way that my statement that your cat is blue and mean depends on the blueness and meanness of your cat. The relevant facts are the inputs, and they are themselves statements that may be either true or false; the statement ’ your cat is blue’ is one imput, and it is either true or (hopefully) false. What a truth table does is list, usually in a systematic fashion, all the possible combinations of values of these inputs. Then, these values are plugged into the logical statement, and the table records whether or not the statement is true in each case. If the statement is simple, such as ‘A and B’, then the table only needs three columns;

A B (A and B)

T T T

T F F

F T F

F F F

If the statement is more complicated, more columns are useful to break down the elements of the statement, although they are not strictly necessary. For example: “A and not (B XOR C)” (XOR means exclusive or; only one or the other)

A B C (B XOR C) (not B XOR C) A and not (B XOR C)

T T T——F——————T—————-T

T F T —-T——————-F—————F

T F F —-F——————T—————-T

T T F——T——————-F—————F

F T T——F——————- T—————F

F F T——T——————- F—————F

F F F——F——————- T—————F

F T F——-T——————- F————-F

So here the first three columns are the possible values of the inputs A, B, and C, the second column is the logical result of these inputs (although it depends only on B and C of course), the third column is simply the negation of that, and the final column is the value of the statement: whether or not A and (not B XOR C). So while you could have simply plugged each combination of values into the initial statement, this might be confusing, and the table breaks it into bite sized chunks.

My apologies if I missed something, or if I totally underestimated your level of understanding. Agin, feel free to ask for further explanation.

Just as a clarification for @IBERnineD , XOR is basically how the english language uses the word “OR.” In English, if you tell someone they can have a piece of cake or a piece of pie, you mean that that can either have cake, or pie, but not both. In mathematics, if you tell someone they can have a piece of cake or a piece of pie, that means they could have a piece of cake, or a piece of pie, or both. which is why I love math. Cake and pie!

That being said, if you are given the option [in math] to have cake xor pie, then you can only have cake, or pie, not both.

My hope is to explain AND well… and for you to fill in the blanks, per se.

Consider these two statements (fill in the blank for the first one)

(A) My favourite food is ________.

(B) I had my favourite food for supper last night.

Now, if (A) is true (which it probably is) then we assign it a value of 1.

if (A) is false, then we assign it a value of 0. The same goes for (B).

**(A) AND (B)**

If (A) and (B) are both true then this statement is true:

*My favourite food is pizza and I had it for supper last night*

If only one of the statements is true and the other false then the statement (A) and (B) is false (because either pizza is not your favourite or you did not have it for supper last night). If both (A) and (B) are false, then, of course, the statement that both are true is definitely false.

This is usually summarized in a table as follows:

|......A…...|......B…...|...A and B….|

|....true….......true….......true….....|

|....false…......true…......false…....|

|....true….......false….....true….....|

|....false…......false….....false…....|

Of course, you won’t usually see “false” and “true”, but 0 and 1, respectivelly. I hope this helps a little bit :-)

M><M.

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