Introduction to Logic and Truth TablesDate: 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? Probably in elementary school you memorized your addition tables (and 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 Written this way, we read it as we read an addition table: we read one 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/ |
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