1. Give a subset that satisfies all thefollowing properties simultaneously:SubspaceConvex setAffine setBalanced setSymmetric setHyperspaceHyperplane2. Give a subset that satisfies some ofthe conditions mentioned in (1) but notall, with examples.3. Provide a mathematical example(not just an explanation) of the union oftwo balanced sets that is not balanced.4. What is the precise mathematicalcondition for the union of twohyperspaces to also be a hyperspace?Provide a proof.edited 9:11

1. Subset Satisfying All Properties Simultaneously A hyperplane through the origin in a vector space…

Answered: 1. Give a subset that satisfies all the…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.4: Spanning Sets And Linear Independence

Problem 25E: Determine whether the set S={1,x2,2+x2} spans P2.

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Determine whether the set S={1,x2,2+x2} spans P2.
Express (AB)(AB) in terms of unions and intersections that involve A,A,B,andB
Here are some set definitions that we will refer to in the exercises of this assignment. A={ 1, 3, 5, 7,9} • B={x| (x ≥ 1) ^ (x ≤ 9) } • C= {x| (x A) ^ (x = B) } • D={x| (vy)(x*y = y)} • E= {x|(3y)(x*y=y)} For each statement below, answer whether the statement is true or false, and briefly explain why. 1. 3€ A 2. 4€ A 3. 3 € B 4. 4 € B 5. 3 € C 6. 4 € C 7. CC B 8. BCC [ S 9. DcA 10. ECD G D C A D = I can give you any inteyen and call it x. Each x I give you has a y, so that for all Values of yx*y = y x=3 x=5 X= 7 X |Y LY|XOY X: = XIXI хочу |-||-|| Te و ال 13 Fol نعاما X= | 3 Since X Y X+Y 3-1 3 3 212 313 e- for all 1 the only this this is 3 bの 6 y x.. 5-4-574 0070 5 171 2 10 72 315 23 کی Š 5 5 хса alm X Y xey 9 T -q 90 00 9 9 18 9222 time the relation is true is when is false as it would need to be true in the set members
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AnotneT set theore 1c concept which iS Usejul in both mat matics and computer science is 'symmetric difference'. It is an amalgamation of union and difference and is defined as follows: The symmetric difference between sets A and B is the set AAB = (A – B)U (В - А). (a) Construct a Venn diagram that expresses the meaning of AAB. ( b) Compare ΑΔΒ to ΒΔΑ 2. UNION, INTERSECTION AND SET DIFFERENCE 25 В аnd B — A. (c) Contrast your finding in part b with a comparison of A (This helps explain what led people to the definition of symmetric differ- ence!)
I provided an example. Please solve the problem accordingly
Explain with an example of a topological space which is singleton that is open but not close and another one that is close but not open
6. The sets A and B are defined as: A = {2,5, 7} B = {*,+} (a) Express the set A x B in roster notation. (b) Give an example of an element in A × B × B × B. (c) Give an example of an element in B³. (d) What is A × (Ø.
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8. Prove that the union of two domains is a domain if and only if they have at least one point in common. Note: you may use the (topological) fact that union of two open sets (in C) is open (in C).
Let X be a nonempty set with subsets, A, B, and C. Prove that A ∩ (B \ C) = (A ∩ B) \ (A ∩ B). I know A \ B = A ∩ B' and I know I got to use that somehow but I don't know how. My professor told us that starting on the right might be easier.
1. Let A = = {1,2} and let B = {x,y}. Write out the following sets by listing out all of their elements. (a) Ax B and B x A (b) P(A) and P(B) (c) P(A) × P(B) (d) P(A x B)

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