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I cannot figure out this last question on my Review packet...
Let A = the set of all non-negative integers.
Let relation R on A be defined by (x,y) in R when (x-y) is divisible by 3.
Discuss whether or not R is reflexive, symmetric, or transitive.
This is the only one I am having trouble with, any help is appreciated!
What exactly are you having trouble with?
Write the definitions of reflexive, symmetric and transitive relations and substitute the definition of R there. Then try proving the resulting statements. Feel free to post any difficulties.
I guess I am not sure where I would get the pairs from. I am confused with (x-y) is divisible by 3. Just wondering how I would get a set of pairs in order to figure out how they are reflexive, and transitive. Those I know how to do. Do i just plug in to number in x and y that are divisible by 3 and go from there? Thank you
I don't know a way of showing what you want without starting with the definitions of reflexive, symmetric and transitive relations for this particular R. Once you do it formally, everything will be clearer and easier.
Well the definition of Reflexive is that all sets are present such as (1,1) (2,2) (3,3) etc...
Symmetric is their reverse such as (1,2) (2,1)
and transitive is (x,y) (y,z) = (x,z)
I guess I am not sure how you would find any of the sets
First, (1,1) and the like are not sets but ordered pairs. If you have two equal sets {a, b} and {a', b'}, each with two elements, it does not follow that a = a' and b = b'. It may be that a = b' and b = a'. However, if ordered pairs (a, b) and (a', b') are equal, then a = a' and b = b'.
Second, real definitions of reflexivity and so on start with "for all". For example, R is called symmetric if for all x, y in A, if (x, y) is in R, then (y, x) is in R. Substituting the definition of R, we get: For all x, y, if x - y is divisible by 3, then y - x is divisible by 3.
The way to prove a statement that starts with "For all" is to say, "Fix arbitrary ...". Here we say "Fix arbitrary x and y", and we have to prove "If x - y is divisible by 3, then y - x is divisible by 3". Next, to prove "If A, then B", one says, "Assume A". So we say, "Assume x - y is divisible by 3". All this is done without thinking because this is the standard way to prove statements of this form.
So, x - y being divisible by 3 means, by definition, that there exists an integer k such that x - y = 3k. You need to show that y - x is divisible by 3, which by definition means that there exists an integer m such that y - x = 3m. Can you find such m?
Reflexivity and transitivity should be approached similarly.
I am not sure Im sorry
A math proof is a game that follows certain rules. (In fact, this is not just a metaphor; this can be given a very precise meaning.) One of the rules is: "To prove a statement 'For all x, ...', start with 'Fix an x.'". Another important kind of rules are definitions. Familiar terms such as "small", "good", "stable", "symmetric", etc. are often used in math, but they don't mean what the dictionary says. Instead, there words have been given precise meanings in definitions. You need to learn the rules of this game before you can construct any proof.