Let S be a set, and R an antisymmetric relation on S. Prove that R^c is trichotomous.
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f
(y). a. Prove that R is an equivalence relation on A. b. Let Ex =
fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x
2 Ag to be the collection of all equivalence classes. Prove that
the function g : A ! E deÖned by g (x) = Ex is...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
4. Let S = {1,2,3). Define a relation R on SxS by (a, b)R(c,d) iff a <c and b <d, where is the usual less or equal to on the integers. a. Prove that R is a partial order. Is R a linear order? b. Draw the poset diagram of R.
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the set S defined as follows: Va,bE S, arb if and only if every prime number that divides a is a factor of b and a S b. The relation T is a partial order relation (you do not need to prove this). Draw the Hasse diagram for T
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the...
Let S = Z and R be the relation defined by R = {Z times Z - (n, n)|n Element Z}. (a) Define the relation R, that is aRb if and only if ..... (b) Prove that R^2 = Z times Z
probelms 9.1
9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
[Partial Orders - Six Easy Pieces] A binary relation is R is said to be antisymmetric if (x,y) ER & (y,x) ER = x=y. For example, the relations on the set of numbers is antisymmetric. Next, R is a partial order if it is reflexive, antisymmetric and transitive. Here are several problems about partial orders. (a) Let Ss{a,b} be a set of strings. Let w denote the length of the string w, i.e. the number of occurrences of letters (a...
13 pts) Let R be the relation on R deÖned by
xRy means "sin2 (x) + cos2 (y) = 1".
Recall the Pythagorean identity: 8u 2 R we have sin2 (u) +
cos2 (u) = 1.
(a) (9 pts) PROVE that R is an equivalence relation on
R.
(b) (4 pts) Describe all elements of the (inÖnite) equivalence
class [0].
Recall: sin(0) = 0 and cos(0) = 1.
2. (13 pts) Let R be the relation on R defined by...
Please answer all parts. Thank you!
20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R × S by (a, s) ~ (a, s') if there exists an s"e S such that s* (s,a-sa,) a. 0. Show that ~ is an equivalence relation on b. Let a/s denote the equivalence class...