show the following set identity, given sets A and C
need some with these. thanks
(a) If E1, E2, En are sets, show rI b) Show that the empty set is a subset of every set c) Show that EnE (d) Show that if E is any event of a sample space S, then E UE -S (e) Show that i E CF, ten F EU(En F). Also show the sets E and En F are disjoint. (1) Show for any two sets, E and F, we have F-(EnF)U(EnF). Also...
6 Show the following set identities, giren sets A, B, C, D.. ...a) ¢-C¢-A) = ADC 6). (A-BUA=A ...) An (B-6) = (ANB)-(ANG) d) (A-B). (B-A) = 0 e) (A-G) N (B-G) = CANB)- & .f) (B-A) (ANB) = 0 9) CAUB)-B = A-CARB) = A-B h) A-(B-C) = (A-B) U CANG) ..) (A-B)-C = A - (BUC). ..;) (A-B) CC-D) = (ANG)-(BUD).
set theory
Let (Aiher and (Biiei be the families of sets with the same index set 1. Show that if ΠǐE/A; C 11ie, Bi, then Ai C Bi for all i e 1.
Let (Aiher and (Biiei be the families of sets with the same index set 1. Show that if ΠǐE/A; C 11ie, Bi, then Ai C Bi for all i e 1.
Exercise 5.2: Identify the identity elements in the following sets. 1) The group of integral polynomials under addition. 2) The group of integral polynomials under multiplication. 3) The set of integral polynomials under composition. 4) The set SL3(Z) (that is, matrix entries are integers). 5) The set SL3(R) (matrix entries are real numbers). 6) The set SL3C) (matrix entries are complex numbers).
1. Given a family A,...An of sets (not necessarily disjoint), a trunsversal is a set T such that T (a, .an) where the a,'s are distinct, and a, E A, for all i. A partial transversal is a transversal for A,…4,2. . . . , Aix for some subfamily of the A's. Show that the family of all partial transversals forms a matroid on the ground set E UA). (Hint: think of bipartite matchings.)
1. Given a family A,...An of...
(3) Determine which of the following sets is linearly independent. 02-1 (a) If the set is linearly dependent, express one vector as a non-zero linear combination of the other vectors in the set. (b) If the set is linearly independent, show that the only linear combination of the above vectors which gives the zero vector is such that all scalars are zero. (c) For each of the sets, determine if the span of the vectors is the whole space, a...
Consider the following examples of a set S and a binary operation on S. Show with proof that the binary operation is indeed a binary operation, whether the binary operation has an identity, whether each element has an inverse, and whether the binary operation is associative. Hence, determine whether the set S is a group under the given binary operation. (f) S quadratic residues in Z101 under multiplication modulo 101
Consider the following examples of a set S and a...
10 Express each of the following sets as a Cartesian product of sets: (a) The set of all possible 3-course meals (entrée, main course and dessert) at a restaurant. (b) The set of car registration plates consisting of three letters followed by three digits. (c) The set of all possible outcomes of an experiment in which a coin is tossed three times.
Given the following sets, find the set AU(BNC). U = {1, 2, 3, ..., 10; A = {1, 2, 3, 4 B = {2,3,4} C = {1, 2, 4, 5, 6) Select the correct choice below and, if necessary, fill in the ans in DA. AU (BOC)= [] (Use a comma to separate answers as needed. Use asc 01 OB. AU(BOC) is the empty set.
.6.29. Show that the following pairs of sets have the same cardinality. a) Integers divisible by 3, and the even positive integers (b) R, and the interval (0, oo). (c) The interval [0,2), and the set [5,6)U7,8) (d) The intervals (-oo,-1) and (-1,0)
.6.29. Show that the following pairs of sets have the same cardinality. a) Integers divisible by 3, and the even positive integers (b) R, and the interval (0, oo). (c) The interval [0,2), and the set [5,6)U7,8)...