[Partial Orders - Six Easy Pieces] A binary relation is R is said to be antisymmetric...
Given the following binary relations: The relation Rl on {w, 1, y, z), where R1 = {(w, w), (w, 1), (x, w), (x, 1 ), (x, z), (y, y), (z,y),(2, 2)). The relation R2 on (a, b, c), where R2 = {(a, a ), (b, b), (c, c), (a, b), (a, c), (c, b)}. The relation R3 on {x,y,z}, where R3 = {(1, 2), (9,2), (2, y)}. Determine whether these relations are: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive?
Math 240 Assignment 4 - due Friday, February 28 each relation R defined on the given set A, determine whether or not it is reflexive, symmetric, anti-symmetric, or transitive. Explain why. (a) A = {0, 1,2,3), R = {(0,0).(0,1),(1,1),(1,2).(2, 2), (2.3)} (b) A = {0, 1,2,3), R = {(0,0).(0,2), (1,1),(1,3), (2,0), (2,2), (3,1),(3,3)} (c) A is the set of all English words. For words a and b, (a,b) E R if and only if a and b have at least...
*ESPECIALLY PART D PLEASE 111111 1. Let R be a relation on RxR defined by (a,b)R(c,d) if and only if a - b = c-d DIDUD a) (5 points) Prove that is an equivalence relation on RxR. b) (5 points) Describe all ordered pairs in the equivalence class of (0,0) c) (5 points) Describe all ordered pairs in the equivalence class of (3,1) d) (5 points) Describe the partition of Rx Rassociated with R.
10. TRUE or FALSE: Write TRUE if the statement is always true; otherwise, write FALSE. _a. {0} c{{0}, {{0}}} _b. Ø $ ({1, 2}), the power set of {1,2} c. If5<3 then 8 is an odd integer. d. The relation R = {(a,b), (b,a)} is symmetric but not transitive on the set X = {a,b}. e. The relation {(1,2), (2,2)} is a function from A={1,2} to B={1,2,3} _f. If the equivalence relation R = {(1,1), (2,2), (3,3), (4,4), (1,3), (3,1),...
2. (5pt) Consider the following binary relations. In each case prove the relation in question is an equivalence relation and describe, in geometric terms, what the equivalence classes are. (a) Si is a binary relation on R2 x R2 defined by z+ly-+ 1 r,y). (,y) e S Recall that R =R x R. (b) Sa is a binary relation on R defined by 1-ye2 r,y) e S
Discrete Mathematics. Let A = {2,3,4,6,8,9,12,18}, and define a relation R on A as ∀x,y ∈ A,xRy ↔ x|y. (a) Is R antisymmetric? Prove, or give a counterexample. (b) Draw the Hasse diagram for R. (c) Find the greatest, least, maximal, and minimal elements of R (if they exist). (d) Find a topological sorting for R that is different from the ≤ relation.
2. A binary string is a finite sequence u-діаг . . . an, where each ai is either 0 or 1. In this case n is the length of the string v. The strings ai, aia2,... ,ai... an-1,ai... an are all prefixes of v. On the set X of all binary strings consider the relations Ri and R2 defined as follows: Ri-(w, v) w and v have the same length ) R2 = {(u, v) I w is a prefix...
2. A binary string is a finite sequence v = a1a2 . . . an, where each ai is either 0 or 1. In this case n is the length of the string v. The strings a1, a1a2, . . . , a1 . . . an−1, a1 . . . an are all prefixes of v. On the set X of all binary strings consider the relations R1 and R2 defined as follows: R1 = {(w, v) | w...
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,...
(17) (20pt) Let F be the set of functions f : R+ → R. Prove that the binary relation "f is 0(g)" on F is: (a) (4pt) Write down the definition for "f is O(g)". (b) (4pt) Prove that the relation is reflexive (c) (6pt) Prove that the relation is not symmetric. (d) (6pt) Prove that the relation is transitive. (17) (20pt) Let F be the set of functions f : R+ → R. Prove that the binary relation "f...