Question

Exercise 4.11. For each of the following, state whether it is true or false. If true,...

Exercise 4.11. For each of the following, state whether it is true or false. If true, prove. If false, provide a counterexample.

(i) Let X and Y be sets from Rn. If X ⊂ Y then X is closed if and only if Y is closed.

(ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex then either X or Y is closed and convex (one or the other).

(iii) Let X be an open set and Y ⊆ X. If Y ≠ ∅, then Y is a convex set.

(iv) Suppose X is an open set and Y is a convex set. If X ∩ Y ⊂ X then X ∪ Y is open.

(v) If A and B are closed sets, and A ∩ B = ∅ then A ∪ B is a closed set.

(vi) If a set A is closed, then A cannot be open.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Exercise 4.11. For each of the following, state whether it is true or false. If true,...
Exercise 4.11. For each of the following, state whether it is true or false. If true, prove. If false, provide a counterexample. (i) LetX andY besetsfromRn. IfX⊂Y thenX is closed if and only if Y is closed. (ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex then eitherX or Y is closed and convex (one or the other). (iii) LetX beanopensetandY ⊆X. IfY ≠∅,thenY isaconvexset. (iv) SupposeX isanopensetandY isaconvexset. IfX∩Y ⊂X then X∪Y...
True or False: Any finite set of real numbers is complete. Either prove or provide a...
True or False: Any finite set of real numbers is complete. Either prove or provide a counterexample.
(a) Let the statement, ∀x∈R,∃y∈R G(x,y), be true for predicate G(x,y). For each of the following...
(a) Let the statement, ∀x∈R,∃y∈R G(x,y), be true for predicate G(x,y). For each of the following statements, decide if the statement is certainly true, certainly false,or possibly true, and justify your solution. 1 (i) G(3,4) (ii) ∀x∈RG(x,3) (iii) ∃y G(3,y) (iv) ∀y¬G(3,y)(v)∃x G(x,4)
Is the following statement true? "If f (Y − X) = f (Y ) − f...
Is the following statement true? "If f (Y − X) = f (Y ) − f (X) for all sets X and Y with X ⊆ Y ⊆ A, then f : A → B is injective." Please provide a proof if it is true, and a counterexample if it is false.
1. In this problem, the domain of x is integers. For each of the statements, indicate...
1. In this problem, the domain of x is integers. For each of the statements, indicate whether it is TRUE or FALSE then write its negation and simplify it to the point that no ¬ symbol occurs in any of the statements (you may, however, use binary symbols such as ’̸=’ and <). i. ∀x(x+ 2 ≠ x+3) ii. ∃x(2x = 3x) iii. ∃x(x^2 = x) iv. ∀x(x^2 > 0) v. ∃x(x^2 > 0) 2. Let A = {7,11,15}, B...
Given that A, B, and C are sets, determine if each statement below is true or...
Given that A, B, and C are sets, determine if each statement below is true or false. Prove your answer using set builder notation and logical equivalences and/or giving a counterexample. i. If A ⋃ C = B ⋃ C, then A = B. ii. If A = B ⋃ C, then (A − C) ⋃ (B ∩ C) = B
1) For each of the following state whether it is true or false. If true give...
1) For each of the following state whether it is true or false. If true give one sentence exolanation. If false, give counterexample. a)If f is differentiable on (0,1) and f is increasing on (0,1) then f' > 0 on (0,1) b)Suppose that f is thrice differentiable function defined on (-1,1). Suppose that the second order Taylor polynomial of f at 0 is 1-x^2. Then f has a local extremum at 0.
For Problems #5 – #9, you willl either be asked to prove a statement or disprove...
For Problems #5 – #9, you willl either be asked to prove a statement or disprove a statement, or decide if a statement is true or false, then prove or disprove the statement. Prove statements using only the definitions. DO NOT use any set identities or any prior results whatsoever. Disprove false statements by giving counterexample and explaining precisely why your counterexample disproves the claim. ********************************************************************************************************* (5) (12pts) Consider the < relation defined on R as usual, where x <...
3. For each of the following statements, either provide a short proof that it is true...
3. For each of the following statements, either provide a short proof that it is true (or appeal to the definition) or provide a counterexample showing that it is false. (e) Any set containing the zero vector is linearly dependent. (f) Subsets of linearly dependent sets are linearly dependent. (g) Subsets of linearly independent sets are linearly independent. (h) The rank of a matrix is equal to the number of its nonzero columns.
Identify the false statements. I. The pension adjustment for a Defined Contribution Pension Plan is calculated...
Identify the false statements. I. The pension adjustment for a Defined Contribution Pension Plan is calculated as [9 x Earnings Rate x Pensionable Earnings] - $600. II. An unmarried recipient of a corporate pension plan can choose a guaranteed period, with any remaining payments in the period paid to the beneficiary’s estate should he/she die before the period is up. III. RRIF withdrawals qualify as pension income that can be split for tax purposes between spouses younger than 65. IV....
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT