2) Let X = {ai, a2. аз-G4.a5} be a set equipped with two binary relations *1 and #2 with the following tables 2a12345 (a) Are *1 and *2 binary operations? (b Are both of these relations associative?...
1. Let G = {a, b, c, d, e} be a set with an associative binary operation multiplication such that ab = ba = d, ed = de = c. Prove that G under this multiplication cannot consist of a group. Hint: Assume that G under this operation does consist of a group. Try to complete the multiplication table and deduce a contradiction. 2. Let G be a group containing 4 elements a, b, c, and d. Under the group...
let g=(x e R:x>1) be the set of all real numbers greater than 1. for X,Y e G, define x * y=xy - x-y +2. 1. show that the operation * is closed on G. 2. show that the associative law holds for *. 3.show that 2 is the identity element for the operation *. 4. show for each element a e G there exists an inverse a-1 e G.
2.) Let Z the set of integers and two binary operations on it: Z23(x,y) → xTy = xy + 3x +3y +6 e Z i) Show (Z,L,T)is an integral domain ii) Find the set of units U(Z) 2.) Let Z the set of integers and two binary operations on it: Z23(x,y) → xTy = xy + 3x +3y +6 e Z i) Show (Z,L,T)is an integral domain ii) Find the set of units U(Z)
Question 2. Recall that a monoid is a set M together with a binary op- eration (r,y) →エ. y from M × M to M, and a unit element e E/, such that: . the operation is associative: for all x, y, z E M we have (z-y): z = the unit element satisfies the left identity axiom: for all r E M we have the unit element satisfies the right identity axiom: for all a EM we Let K...
Let S = {x ER:[x]<1}=(-1,1). We will refer to E as hyperbolic relativity space. Now a+b define a binary operation by: if a,beR and ab +-1, then aob= 1+ ab Proposition 1. (5,0) is a group. Remark. This is the kind of problem that every student should become competent at doing. Perhaps some of the details here are more challenging than normally but understanding what are the steps to follow in such a problem is basic, and everyone should understand...
Let p(x) = 24 + 23 +1€ Z2[2] and let a = [z] in the field E = Z2[z]/(p(x)), so a is a root of p(x). (a) (15 points) Write the following elements of E in the form aa+ba+ca+d, with a,b,c,d € Z2. i. a“, a, a6, and a 10 ii. a5 +a+ + a2 + 1 iii. (a? + 1)4 (b) (5 points) The set of units E* = E-{0} of the field E is a group of order...
9. (a) Let Ao(x) = / (1-t*)dt, Ai(z) = / (1-t2) dt, and A2(z) = / (1-t2)dt. Compute these explicitly in terms ofェusing Part 2 of the Fundamental Theorem of Calculus. b) Over the interval [0,2], use your answers in part (a) to sketch the graphs of y Ao(x), y A1(x), and y A2(x) on the same set of axes. (c) How are the three graphs in part (a) related to each other? In particular, what does Part 1 of...
2. A binary string s a finite sequence u = ala2 . . . an, where each ai įs either 0 or 1. In this case n is the length of the string v. The strings ai,aia2,...,ai...an-1,aan are all prefixes of v. On the set X of all binary strings consider the relations Ri and R2 defined as follows R, = {(u, u) | w and u have the same length } {(w, u) | w is a prefix of...
Problem 4.22. Five basic properties of binary relations R : A-> B are: 1. R is a surjection1 in 2. R is an injection [S 1 in] 3. R is a function 1 out 4. R is total1 out] 5. R is empty0 out] Below are some assertions about R. For each assertion, indicate all the properties above that the relation R must have. For example, the first assertion impllies that R is a total surjection. Variables a, a1.... range...
QUESTION 10 The equality relationon any set S is: A total ordering and a function with an inverse. An equivalence relation and also function with an inverse. A function with an inverse, and an equivalence relation with as single equivalence class equal to S An equivalence relation and also a total ordering QUESTION 11 A binary operation on a set S, takes any two elements a,b E S and produces another element c e S. Examples of binary operations include...