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Introduction to Logic and Truth Tables

```
Date: 09/27/2000 at 18:49:57
From: Carolyn
Subject: Logic

I can't figure out the p and q thing.

I don't understand about ^, v, and stuff like that. Could you explain
it to me in an easy way?
```

```
Date: 10/02/2000 at 14:29:07
From: Doctor TWE
Subject: Re: Logic

Hi Carolyn - thanks for writing to Dr. Math.

The p's and q's represent "propositions." They're kind of like
variables in algebra, only instead of representing numbers, they
represent things that are either true or false. The statement "today
is Monday" is a proposition because it can be either true (if, indeed
it is Monday) or false (if, for example, it's Tuesday.) You can also
think of it as being a variable that can only have one of two values:
0 (for false) or 1 (for true.) In fact, that is exactly how computers
deal with them.

The logic expression p^q (read "p and q") is similar to the algebra
expression A+B. A and B represent numbers, and A+B means we're
combining those numbers in a certain way. If A = 3 and B = 5, then
A+B = 3+5 = 8. How do we know 3+5 = 8? Because somewhere along the
line we defined the addition operation and said that 3+5 = 8.

Similarly, if p = T (true) and q = F (false) then p^q = T^F = F. (Note
that the "variables" p and q take on the "values" T and F instead of
the "values" 3 and 5.) How do we know that T^F = F? Because (just as
with addition) somewhere along the line we defined the AND operation
and said that T^F = F.

How do you know all the possibilities for addition of numbers?
your subtraction tables, multiplication tables and division tables,
too). Similarly, we have "truth tables" that tell us what the results
of logic operations are. Unlike addition tables, which typically have
100 sums to memorize, 2-variable truth tables have only 4 results for
you to memorize. This is because each proposition (or variable) can
only be one of two values: T or F. Here's the AND truth table:

^ | T F
===+=====
T | T F
F | F F

value across the top to find the column and one down the left side to
find the row. Where that row and column intersect is the answer. So,
for example, for T^F we'll use the first column (labeled T) and the
bottom row (labeled F).

______ Our column
/
|
V
+-+
^ | |T| F
===+=+=+===
T | |T| F
+--+-+-+--+
|F | |F| F| <- Our row
+--+-+-+--+

Where that row and column intersect we read our result: F.

More often, however, logic truth tables are written with each
proposition (or variable) in a column and the result in a separate
column. Here's the more common way to write the AND truth table:

p q | p^q
-----+-----
T T |  T
T F |  F
F T |  F
F F |  F

In this type of truth table, we look for the row that has the correct
values for p and q, and read the result from the final column. For
example:

p q | p^q
-----+-----
T T |  T
+----+----+
|T F |  F | <- Our row
+----+----+
F T |  F
F F |  F

so T^F = F.

Just as arithmetic starts off with a few simple operations (addition
and subtraction), then builds to more complex operations (division,
multiplication, exponents, etc.), the same holds for logic operations.
The two basic operations are AND (p^q) and OR (pvq). From these, we
can build more complex operations like implication, equivalence,
exclusive-or, and so on. If you want to know more about these
secondary operations, write back.

I hope this helps.

- Doctor TWE, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Logic
Middle School Logic

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