Consider the set Z, × Z, = {(x, y) : x € Z, and y € Z3}. Define the operation/sum O in Z2 × Z3 such that for any two elements (a, b) and (c, d) in Z, x Z3, we have (a, b) (c, d) = (a +2 c, b +3 d) where +2 is the addition modulo 2 in Z2, and +2 is the addition modulo 3 in Z3. 1. a. List the element of Z, x Z3. b. Determine the identity element in Z, × Z3. c. Determine the inverse element of each element in Z, x Z3. d. Is Z, x Zz a group under the given operation? Why? Is it Abelian? e. List a proper subgroup of Z, × Z3. Write "none" if there are no proper subgroups.

Q: Find the number of elements in the conjugacy class of (abc)(def) in S₆

A: We have to find the number of elements in the conjugacy class of (a b c)(d e f) in S₆

Q: Let A, B, C, D be sets. Prove that (a) A × (BN C) = (A × B) N (A × C') (b) (A × B) N (C × D) = (AnC)…

A: Given (a) A×B∩C=A×B∩A×C (b) A×B∩C×D=A∩C×B∩D

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A: Consider the given information:Let a,b,x,y∈ℤ be such that ax+ by=11.

Q: Let A= {1,2,x,y,z}, and B= {1,2,3,z}. How many distinct elements are in (A*B) intersect (B*A

A: Given that A=1,2,x,y,zB=1,2,3,z To find the number of distinct elements in A×B∩B×A:

Q: find the amount of odd three-digit numbers that can be created. (

A: here given digits are  [0, 1, 2, 3, 4, 5, 6]. so there are total 7 digits . here use counting…

Q: Let a, b be nonzero integers. Prove that (a, b) = 1 if and only if (a + b, ab) = 1.

A: Given  Let a, b be nonzero integers  To Prove that gcd (a, b) = 1 if and only if gcd(a + b, ab) = 1…

Q: Let [A1, A2], [B1, B2], [C1, C2] be 3 married couples who attended a concert.  In how many ways can…

A: Permutation: A permutation is an arrangement of objects in a definite order. Let n is the total…

Q: et z = 6ex²y³ find O 12xy³ex²y³ O 18x²y²ex²y³ O 18xy³x²y³ O 12x²y²ex²²³ дz ax

A: Given, z=6ex2y3

Q: Let X={a,b,c,d), f = ((a,b),(b,a),(c,c),(d,d)}CX×X, and g= ((a,d),(b,a),(c,b),(d,c)}CX×X. Find gof.…

A: Use the definition of composition of functions to solve this problem as in next step.

Q: In cartesian product, prove that (A-B)XC=(AXC)-(BXC) A,B,C are sets

A:

Q: Define two operations + and * on A = Z × Z as follows: for (a, b) and (c, d) in A: ɔy (a, b)+(c, d)…

A: As per guidelines, we will solve first three subparts: Define two operations + and * on A=ℤ×ℤ as…

Q: Let w be defined as 2 more than the number of digits in the integer w. For example, 15 =4(2 digits…

A:

Q: Let a, b € N. Choose all correct statements. O gcd(-a, b) = gcd(a, -b) O ged(2a – b, a + b) =…

A:

Q: Suppose that y = y(x). Find y'(x) by implicit differentiation x*y +x sin y = 4 VI +y = 1+x² + y? y…

A: " Since you have asked multiple questions, we will solve the first question for you. To get the…

Q: Let V4 = 4 V2 = V3 = -2 Pi -2 P2 2 2 and S = {v1, v2, V3, V4}. Determine whether pl and p2 are in…

A:

Q: Suppose the multiplication cx is defined to produce (cx1, 0) instead of (cx1, cx2 ). With the usual…

A: The multiplication cx where x∈ℝ2 is defined as cx=cx1, 0 because x=x1, x2 where x1 and x2 belong to…

Q: A cosets of of 212 are اخترأحد الخیارات + 2, +1,.1 O + 1,.2 O + 4, + 1, .3 + 4, + 2, + 1 , .4 O

A: Cosets Definition: Let G be a group and let H is a nonempty subset of G. A left coset of H in G is a…

Q: Show that there is exactly one positive number u such that In(u² + 1) + Tu = 2021.

A:

Q: nsist only of of even length. Generalize to find the EGF number of ations that consist only of…

A: To find: (1) The EGF for the number of permutations that consist only of cycles of even length.…

Q: Let f and g be permutations. Show that the order of h=fgf^−1 is the same as the order of g.

A:

Q: Let V be all elements of R2 that have the form (a, a2) where a is a real number. Show that (3,9) is…

A: Let V be all elements of R2 that have the form (a, a2) where a is a real number. We have show that…

Q: Let A = { {a}, {b}, {c}, {d}, {e} } Determine by extension P(A

A:

Q: Let A be the set of all positive integers whose last two digits form a number that is evenly…

A: We have the given condition as   Let us consider A be the set of all positive integers whose last…

Q: Let a, b be nonzero integers with (a, b) = 1. Compute (a + b, a – b).

A:

Q: d = (a, b, c).

A:

Q: Show that A × B ≠ B × A, when A and B are nonempty, unless A = B

A:

Q: Show that the following statement is false. If A and B are two sets of positive integers, then…

A: Let A={2,3,4}    B={5,6,7} U={1,2,3,4,5,6,7} Now, A'={ 1,5,6,7}  B' ={1,2,3,4} Now,…

Q: State the scope of each quantifier in the following expression: "Vy, [Q(x, y) ^ 3r (R(x, y) ^ ~P(x,…

A: The quantifiers used in the expression are ∀,Q, R, ∃, ~P. The scope of the quantifiers are given…

Q: Let a, b, c, d, e, and f be positive integers. Let d = gcd(a, b, c), e = gcd(a, b) and f = gcd(e,…

A: Given that for positive integers a,b,c,d,e,f , d=gcda,b,ce=gcda,bf=gcde,c

Q: Give an example of a closed subset S of R2 such that conv S is not closed.

A:

Q: Let a, b e N. Choose all correct statements. O gcd(a, b) = gcd(-a, -b) O gcd(a, b) = gcd(a², b²) O…

A: We apply property of GCD to solve the problem

Q: Find all of the units in Pn and verify where Pn is the power set of the set {1,2,...,n} with…

A: Find all of the units in  and verify where is the power set of the set  with addition defined as…