24. Let G = < C. Define 0: - by o(n) =i" Verify that o is a homomorphism a) b) Find Ker( o )

Q: Let f= (1436578) and g = (1 825 3) (47) be permutations in Sg, written in cycle notation. What is…

A: This is a problem of abstract algebra.

Q: (a) classes of X induced by Prove that - is an equivalence relation on X and describe the set X/ ~…

A: This is a problem of relation.

Q: Prove that the structure (even integers, +, *) is closed with respect to *.

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Q: Define relation R on (Z+ U {0}) × Z+ by (a, b)R(c,d) if ad = bc. Describe the equivalence class [(2,…

A: Our objective is given below:

Q: The relation R defined on a set A is antisymmetric if (a, b) ER = (b, a) E R for (A) Every (a, b) ER…

A: A relation R is said to be antisymmetric if (a,b)∈R and (b,a)∈R implies a=b

Q: Find example where of is continuous and an - bijective but f¹ is not

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Q: Consider the relation being described by the following arrow diagram. Which of the these statements…

A: (i) It is given that ;      S = {6, 7, 8}  and  T = {2, 4, 6} And the relation between S and T is :…

Q: Dijec not need to prove you function is a bijection): (a) N and Z (b) Rand (1, ∞) (c) (0, 2) and (3,…

A:

Q: is ----Llarder for - 46 8. The Ker() for the homomorphism :Z-Z₁, such that (1)=9is r 6, and r **…

A:

Q: b) A function g: Z → Zis defined by the rule g(m, n) = m + n Determine whether it is onto or not..

A: We will answer the first question as we don't answer multiple questions at a time. Please resubmit…

Q: let N be the set of natural numbers and f:NU {0} →NU [0] be definedby: (n+1 when n is even f(n) =…

A: Prove that the function f is injective for x∈N. \ The function is injective as the function is…

Q: 20 - If f is a function on a set A= {1,2,3,4,5} such that f= { (1,2), (2,3), (3,4), (4,x), (5,5)}.…

A: Use the property that in bijective functions each element of doamin and codomain has image and pre…

Q: Define a bijective map g : x2/3--->R.

A: Sol

Q: Let A = {2" : m e Z}. A relation - is defined on the set Q+ of positive rational nu a a ~ b E A is…

A:

Q: Show that there exist only two homomorphism from R into R?

A:

Q: Let f R2 R² be an isomorphism where : → ƒ ( [¹³]) = [()] and ƒ ([×4]) = }}] Find(G)

A:

Q: 2. Consider morphisms f: A → B, g: BC, h: CD, k: DB, j: BC, t: A → C in a category C so that g is an…

A: Given: Morphisms f:A→B, g:B→C, h:C→D, k:D→B, j:B→C and l:A→C in a category C, where g is an…

Q: Prove relation B is an equivalence relation

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Q: The relation by a = b (modn) is Partial Order Relation Ture O False O

A:

Q: Prove that gcd(a, b) is the least natural number representable in the form ax + by with x, y ∈ Z.

A: We have to show that gcd(a, b) is the least natural number representable in the form ax + by with x,…

Q: 3. A = {2, 2+2, 2+4,2+6, 2+8} where A = {2,4,6,8} 4. A=(1-22, 1+22) where A = Z*

A: Note : (i) ∩α∈S Aα=x : ∀α∈S ,x ∈Aα               (ii) ∪α∈S Aα=x : ∃α∈S ,x ∈Aα

Q: Let R = %3 { la, b e z}and let p:R - Zbe defined by : 0(1 ) = - . 1) is a ring a) Homomorphism. b)…

A: The solution is given by using definitions of homomorphism, isomorphism and kernel as follows

Q: 5) Let R be a binary relation over the positive integers defined as follows: R = {(a, b) | 2b < a <…

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Q: Prove that homotopy with respect to a domain 2 is a transitive property.

A: We will show that homotopy w.r.to the given domain is a transitive property.

Q: 15. Let x = (x1, x2). Using the feature mapping show that *(2,3)·¤((4,4) –-(2,3)·(4,4)*

A: Given that for x=x1,x2, the feature mapping is…

Q: 13. (9 points) Let D be the set of finite subsets of positive integers. Let S be the set of all…

A: The objective of this question is to find the set of all even factors of given numbers using the…

Q: Prove that the mapping f : U(16) ---> U(16) given by f(x) = x^5 is an automorphism.

A: The given map is f:U16→U16 defined by fx=x5. We have to prove that f is an automorphism. We know…

Q: Suppose that we are interested to count the number of integers not divisible by 7, 8, 9 between 1 to…

A: To find the integers between 1 to 10000 that are not divisible by 7,8 and 9. Firstly, find the L.C.M…

Q: How many distinct equivalence classes exist in the relation R defined as below: x Ry + 3| (2x - )

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Q: R = {(x, e¹x) : x = R, n = Z}.

A: A relation is called an equivalence relation if it is reflexive, symmetric and transitive

Q: Find an orthogonal set with the same span (and same number of elements) as [2] [8]} (18118)} "

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Q: 4. In the ficld (C,+, ) of complex numbers, define the mapping f: C- C by the rule f(a, b) - (a,…

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Q: The population of a town grows at a rate proportional to the population present at time t. The…

A: