1. Suppose that G is a group. Prove the following two
statements:
a. If x 2 G then the inverse of x is unique.
b. If x; y; z 2 G then (xyz)?1 = z?1y?1x?1.
e is the identity of group G
1. Suppose that G is a group. Prove the following two statements: a. If x 2...
Theorem 7.5 Let G be a group. (1) G has a unique identity element (2) Cancellation Laws. For all a, b,ce G, if ab ac, then b-c. For all a, b,c E G, if ba-ca, then (3) Each element of G has a unique inverse: For each a E G, there exists a unique element d e G such that ad-e and da e . Prove that each element of a finite group G appears exactly once in each row...
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
(8)5. Do either (a) or (b) but not both. (a) Prove that in a group G, the identity element is unique. GER Lt A&G Let A o I- Afa o.cea.o a. AI .c,Le A (b) Prove that each element in a group G has a unique inverse (do not use the Cancellation Law).
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
• You should be able to prove the following statements: - If y : G → J be a group homomorphism andrEG has finite order, then ord(r) is divisible by ord(y(a)). If p: G → J is a group isomorphism and r e G, then r has infinite order if and only if ø(r) has infinite order. If G, J and G2 J2, then G1 x G2 J x J. If G is a cyclic group and K is subgroup...
Topology 3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
the following questions are relative,please solve them, thanks! 4. Let G be a group. An isomorphism : G G is called an automorphism of G. (a) Prove that the set, Aut(G), of all automorphisms of G forms a group under composition. (b) Let g E G. Show that the map ф9: G-+ G given by c%(z)-gZg", įs an automorphism. These are called the inner automorphisms of G (c) Show that the set of all g E G such that Og-Pe...
4. (4 points) Prove the truth or falsity of the following statements. To prove a statement true, give a formal argument (in cases involving implications among FD's, use Armstrong's Axiom System). To prove falsity, give a counterexample. 1. {A + B, DB → C} F{A+C} 2. {X+W, WZ+Y} F{XZ → WY} 3. {A D, B7C, F + B, CD + E|| F{AF → E} 4. Suppose R is a relation scheme and F a set of functional dependencies applicable to...
Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) = E(X)+ E(Y). (2) Prove Var(X + Y) = Var(X) + Var(Y)2Cov(X, Y). (3) Prove Cov(X, Y) E(XY)- E(X)E(Y). (4) Prove that if X and Y are independent, i.e., f(x, y) Cov(X, Y) 0. Is the reverse true? (5) Prove Cov (aX b,cY + d) = acCov(X, Y). (6) Prove Cov(X, X) = Var(X) fx (x)fy(y) for any (x,y), then =
1. Express in the language of the FOL the following mathematical statement: If S is a set of elements and * is a binary operation in S for which the following four assumptions hold, where = is an equivalence relation: a. S is closed under * ( if x and y are in S then x * y is also in S) b. * is associative (x, y, and z in S (x * y) * z = x *...