Binary Relation Review Questions

Transcription

1 CSE 191 Discrete Structures Fall 2016 Recursive sets Binary Relation Review Questions 1. For each of the relations below, answer the following three questions: Is the relation reflexive? Is the relation symmetric, anti-symmetric or neither? Is the relation transitive? Is the relation a partial order? Is the relation a total order? Is the relation an equivalence relation? If the relation does not have one of the properties, give an example illustrating this or explain why. 1(a). The domain is the set of UB students. xry if and only if person x is taller than person y. Is the relation reflexive? No. A person is not taller than him/herself. If person a is taller than person b, then person b is not taller than person a. Yes, anti-symmetric. There are no times where arb and bra so anti-symmetry is not violated. Is the relation transitive? Yes. If a is taller than b and b is taller than c then a must be taller than c. Is the relation a partial order? No. The relation is not reflexive. Is the relation an equivalence relation? No. Not reflexive or symmetric. 1(b). The domain is the set of UB students. xry is in the relation if person x is a first cousin of person y (i.e., a parent of person x is a sibling of a parent of person y). Is the relation reflexive? No. Your parents are not a sibling of themselves.

2 Is the relation symmetric, anti-symmetric or neither? Yes, symmetric. If x has a parent who is a sibling of a parent of person y, then the opposite is true. No, not anti-symmetric. Any two people who are first cousins violate this rule. Is the relation transitive? No. If you have two different people x and y where x is a first cousin of y and y is a first cousin of x, is is never the case that x is a first cousin of x. Is the relation a partial order? No. The relation is not transitive. Is the relation an equivalence relation? No. Not reflexive or transitive. 1(c). The domain is a set of UB students. xry is in the relation if person x knows the student ID number for person y. You can assume that every student knows his or her own student ID number. Is the relation reflexive? Yes. Everyone knows their own person number (in an ideal world). If I tell everyone my person number that doesn t mean they will tell me their person number. No, not anti-symmetric. It could be the case that two people share their person number with one another. Is the relation transitive? No. If you have three different people a, b, c and a knows b s person number and b knows c s person number, it s possible that a doesn t even know person c so they wouldn t know their person number either. Is the relation a partial order? No. The relation is not antisymmetric or transitive. Is the relation an equivalence relation? No. Not symmetric or transitive. 1(d). The domain is the set of real numbers. xry if and only if x + y = 0. Is the relation reflexive? No

3 Is the relation symmetric, anti-symmetric or neither? Yes, symmetric. If x + y = 0 then y + x = 0. No, not anti-symmetric. 5+( 5) = 0 and ( 5)+5 = 0 but 5 5. Is the relation transitive? No. 5 + ( 5) = 0 and ( 5) + 5 = 0 but Is the relation a partial order? No. The relation is not reflexive, anti-symmetric, or transitive. Is the relation an equivalence relation? No. Not reflexive or transitive. 1(e). The domain is the set of real numbers. xry if and only if x = 2y. Is the relation reflexive? No. 2 2(2) = 4. 2 = 2(1) but 1 2(2). Yes, anti-symmetric. The only time that x = 2y and y = 2(x) are both true is when x = y = 0. Is the relation transitive? No. 4 = 2(2) and 2 = 2(1) but 4 2(1) = 2 Is the relation a partial order? No. The relation is not reflexive or transitive. Is the relation an equivalence relation? No. Not reflexive or symmetric. 1(f). The domain is the set of real numbers. xry if and only if x y is a rational number. Is the relation reflexive? Yes. x x = 0 for all real numbers and 0 is rational. Is the relation symmetric, anti-symmetric or neither? Yes, symmetric. If x y is rational, then y x is that rational number multiplied by 1, which would still be rational. No, not anti-symmetric. 5 3 = 2 is rational and 3 5 = 2 is rational but 5 3.

4 Is the relation transitive? Yes. If x y is rational and y z is rational then x z must also be rational. Is the relation a partial order? No. The relation is not antisymmetric. Is the relation an equivalence relation? Yes. The relation is reflexive, symmetric, and transitive. 1(g). The domain is the set of real numbers. xry if and only if x y is a non-negative number. Is the relation reflexive? Yes. x x = 0 is always non-negative. Is the relation symmetric, anti-symmetric or neither? Not symmetric. 5 3 = 2 is non-negative but 3 5 = 2 is not non-negative. Yes, anti-symmetric. The only time that x y and y x are both non-negative is when x = y. Is the relation transitive? Yes. If x y is non-negative and y z is non-negative then x z must be non-negative. Is the relation a partial order? Yes. The relation is reflexive, antisymmetric, and transitive. Is the relation a total order? Yes. It is a total order since any two real numbers can be subtracted from one another and their difference will either be negative or non-negative. Is the relation an equivalence relation? No. Not symmetric. 1(h). The domain is A = {a, b, c, d}. The relation is {(a, b), (a, a), (b, b), (b, a), (c, d), (d, c)}. Is the relation reflexive? No. d is not related to d. Is the relation symmetric, anti-symmetric or neither? Yes, symmetric. For each pair (x, y) in the relation, we always have (y, x). No, not anti-symmetric. a is related to b and b is related to a but a b. Is the relation transitive? No. c is related to d and d is related to c but c is not related to c. Is the relation a partial order? No. The relation is not reflexive, anti-symmetric, or transitive.

5 Is the relation an equivalence relation? No. Not reflexive.