Hi! I've learned about relations, and about how they can be reflexive, symmetric, or transitive.
I'm unsure of how to prove if a given relation is or is not each of these. Can a relation be more than one of these? Can it be all of them?
Below is a question where I have been asked to identify if each of these relations is reflexive, symmetric, or transitive, and to give a reason why:
1. In the real numbers, x~y if 3x+5y is not rational.
2. In the integers, x~y if 3x+5y is divisible by 8
What is your book's specific definitions of "reflexive", "symmetric", and "transitive"? In mathematics, definitions are everything, so you'll need to be certain and clear on these definitions. (You can review many articles with definitions and worked examples
here. For instance, the definitions with worked examples [both "yes, it is" and "no, it isn't"] may be found
here.)
Looking at the first relation, we have:
1. Let x and y be real numbers. These numbers x and y are related according to the rule "3x + 5y is not a rational number".
a. To be reflexive, it must be true that, for any real number x, this number is related to itself under the relation "3x + 5x = 8x is not a rational number".
Is this true for all real numbers? Can you find a real number such that 8x is irrational? (Think about multiples of irrational numbers like "pi".) So is this relation reflexive? (Note: Only one counter-example is necessary in order to say "no, it isn't.")
b. To be symmetric, it must be true that, for any x and y such 3x + 5y is irrational, then 3y + 5x is also irrational. Is there a counter-example for this? Or can you prove (in generality) that 3y + 5x must also be irrational?
c. To be transitive, it must be true that, for any x and y such that 3x + 5y is irrational, and for any z such that 3y + 5z is irrational, it is also true that 3x + 5z is irrational. Can you use the fact that the middle variable, the y, is not present in the final "then" part, to create a counter-example?
2. Let x and y be integers. These integers x and y are related according to the rule "3x + 5y is divisible by 8".
a. To be reflexive, it must be true that, for any integer x, 3x + 5x = 8x is divisible by 8. Since the expression is an integral multiple of 8, then this is clearly fulfilled by any integer. So... what can you conclude?
b. To be symmetric, it must be true that, for any integers x and y for which 3x + 5y is a multiple of 8, then 3y + 5x is also a multiple of 8. Does this "feel" right to you, or are your instincts leading you to think that this might not be true? Can you find a pair of integers x and y so that 3x + 5y = 8m for some integer m, but 3y + 5x is not? Can you try fiddling with some numbers and see what you come up with? Or can you prove this to be true, in full generality?
c. To be transitive, it must be true that, for any integers x and y for which 3x + 5y is a multiple of 8, and for any z for which 3y + 5z is a multiple of 8, then it must be that 3x + 5z is a multiple of 8. Does this feel right, or should you maybe try for a counter-example?
If I've missed your point, or if you have other questions, kindly please reply with specifics (as, I'm afraid, "I totally don't get it" is not helpful). Thank you!
