Let S be a nonempty set and ⋆ be an operation on S. If ⋆ on...
Let S be a nonempty set and ⋆ be an operation on S. If ⋆ on S is commutative and S′ ⊂ S is closed with respect to ⋆, is ⋆ on S′ commutative? Prove why or why not.
Homework Answers
11*. Suppose S a nonempty subset of a group G. (a) Prove that if S is...
11*. Suppose S a nonempty subset of a group G. (a) Prove that if S is finite and closed under the operation of G then S is a subgroup of G. (b) Give an example where S is closed under the group operation but S is not a subgroup.
Exercise 1.1.2 Let S be an ordered set. Let A CS be a nonempty finite subset....
Exercise 1.1.2 Let S be an ordered set. Let A CS be a nonempty finite subset. Then A is bounded Furthermore, inf A exists and is in A and sup A exists and is in A. Hint: Use induction
Problem 1 [10 points] Prove that if A is a nonempty set of real numbers with...
Problem 1 [10 points] Prove that if A is a nonempty set of real numbers with a lower bound and B is a nonempty subset of A, then inf A <inf B. Problem 2 [10 points) Let A be a nonempty set of real numbers with a lower bound. Prove there exists a sequence (ar) =1 such that are A for all n and we have limntan = inf A.
L maps is a quotient map. 4, Let ( X,T ) be a topological space, let Y be a nonempty set, let f b...
l maps is a quotient map. 4, Let ( X,T ) be a topological space, let Y be a nonempty set, let f be a function that maps X onto Y, let U be the quotient topology on induced by f, and let (Z, V) be a topological space. Prove that a function g:Y Z is continuous if and only if go f XZ is continuous.
l maps is a quotient map. 4, Let ( X,T ) be a topological...
3) Let S be a set with an associative binary operation :SxS->S. Let e, be a...
3) Let S be a set with an associative binary operation :SxS->S. Let e, be a left identity of S (i.e., e, *ssVse S), and let eg be a right identity of S (i.e., a) Prove that e-e b) Also prove that S can have at most one 2-sided identity.
2. Let A be a nonempty set of real numbers bounded above. Define Prove that -A...
2. Let A be a nonempty set of real numbers bounded above. Define Prove that -A is bounded below, and that inf(-A) = -sup(A). -A={-a: aEA . (5 marks) (You may use results proved in class.) A = 0 , A is bounded above.
Real Math Analysis Let A be a nonempty finite subset of R. Prove that A is...
Real Math Analysis Let A be a nonempty finite subset of R. Prove that A is compact. Follow the comment and be serious Please. our goal is to show that we can find a finite subcover in A. However, I got stuck in finding the subcover. It is becasue finite subset means the set is bounded but it doesn't mean it is closed.
(4) Let X be a nonempty set, and let o E Perm(X). The set of fixed...
(4) Let X be a nonempty set, and let o E Perm(X). The set of fixed points of o is fix(0) = {x € X : 0(x) = x}. The support of o is supp(o) = {x EX :0() #x}. (a) Prove that fix(o) U supp(o) = X and fix(on supp(o) = 0. (b) Prove that fix(oº) = fix(0) and supp(02) = supp(o). (c) Permutations o and T in Perm(X) are disjoint if, for all x E X, we have...
1. Let A be a nonempty subset of R such that every number in A is...
1. Let A be a nonempty subset of R such that every number in A
is greater than 2 (NOTE: This doesn’t necessarily mean that A =
(2,∞)).
(a) Explain why A must have an infimum.
(b) Let c = inf(A). Prove that a∈A INTERSECTION (−∞,a] =
(−∞,c].
CAN SOMEONE PLZ HELP ME WITH THIS QUESTION.
1. Let A be a nonempty subset of R such that every number in A is greater than 2 (NOTE: This doesn't necessarily mean...
6. Let F be a field and a Fx] a nonconstant polynomial. Denote (that is, (a(x)) is the set of all polynomials in Flr] which are divisble by a()). Then (a) Prove that (a(x)) is a subgroup of the a...
6. Let F be a field and a Fx] a nonconstant polynomial. Denote (that is, (a(x)) is the set of all polynomials in Flr] which are divisble by a()). Then (a) Prove that (a(x)) is a subgroup of the abelian group (Flx],. (b) consider the operation on F[r]/(a()) given by Prove that this operation is well-defined. (c) Prove that the quotient F]/(a(x) is a commutative ing with identity (d) What happens if the polynoial a() is constant?
6. Let F...
11*. Suppose S a nonempty subset of a group G. (a) Prove that if S is finite and closed under the operation of G then S is a subgroup of G. (b) Give an example where S is closed under the group operation but S is not a subgroup.
Exercise 1.1.2 Let S be an ordered set. Let A CS be a nonempty finite subset. Then A is bounded Furthermore, inf A exists and is in A and sup A exists and is in A. Hint: Use induction
Problem 1 [10 points] Prove that if A is a nonempty set of real numbers with a lower bound and B is a nonempty subset of A, then inf A <inf B. Problem 2 [10 points) Let A be a nonempty set of real numbers with a lower bound. Prove there exists a sequence (ar) =1 such that are A for all n and we have limntan = inf A.
l maps is a quotient map. 4, Let ( X,T ) be a topological space, let Y be a nonempty set, let f be a function that maps X onto Y, let U be the quotient topology on induced by f, and let (Z, V) be a topological space. Prove that a function g:Y Z is continuous if and only if go f XZ is continuous.
l maps is a quotient map. 4, Let ( X,T ) be a topological...
3) Let S be a set with an associative binary operation :SxS->S. Let e, be a left identity of S (i.e., e, *ssVse S), and let eg be a right identity of S (i.e., a) Prove that e-e b) Also prove that S can have at most one 2-sided identity.
2. Let A be a nonempty set of real numbers bounded above. Define Prove that -A is bounded below, and that inf(-A) = -sup(A). -A={-a: aEA . (5 marks) (You may use results proved in class.) A = 0 , A is bounded above.
(4) Let X be a nonempty set, and let o E Perm(X). The set of fixed points of o is fix(0) = {x € X : 0(x) = x}. The support of o is supp(o) = {x EX :0() #x}. (a) Prove that fix(o) U supp(o) = X and fix(on supp(o) = 0. (b) Prove that fix(oº) = fix(0) and supp(02) = supp(o). (c) Permutations o and T in Perm(X) are disjoint if, for all x E X, we have...
1. Let A be a nonempty subset of R such that every number in A
is greater than 2 (NOTE: This doesn’t necessarily mean that A =
(2,∞)).
(a) Explain why A must have an infimum.
(b) Let c = inf(A). Prove that a∈A INTERSECTION (−∞,a] =
(−∞,c].
CAN SOMEONE PLZ HELP ME WITH THIS QUESTION.
1. Let A be a nonempty subset of R such that every number in A is greater than 2 (NOTE: This doesn't necessarily mean...
6. Let F be a field and a Fx] a nonconstant polynomial. Denote (that is, (a(x)) is the set of all polynomials in Flr] which are divisble by a()). Then (a) Prove that (a(x)) is a subgroup of the abelian group (Flx],. (b) consider the operation on F[r]/(a()) given by Prove that this operation is well-defined. (c) Prove that the quotient F]/(a(x) is a commutative ing with identity (d) What happens if the polynoial a() is constant?
6. Let F...
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