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Exercise 3. (i) Use the definition (of a maximum element of a set) to prove that...
12. Definition : Let Λ be a non-empty set. If for each a є Л there is a set Aa, the collection (A...
12. Definition : Let Λ be a non-empty set. If for each a є Л there is a set Aa, the collection (Aa : α Ε Λ is called an indexed collection of sets. The set A is called the index set. Traditionally Λ is often the natural numbers-you are probably pretty used to seeing sets indexed by the natural numbers but it can in fact be any other set! Here's the exercise: Let Л-R+ (meaning the positive real numbers,...
1. [15 pts] Use Definition 1.5 (definition of probability function) to prove Propo- sition 1.3 ()...
1. [15 pts] Use Definition 1.5 (definition of probability function) to prove Propo- sition 1.3 () 15 pts) & (iv) [10 pts). You do not need to prove (i) and (ii). [Definition 1.5/ Let Ω be a set of all possible events. A probability function P : Ω → 0,11 satisfies the follouing three conditions (i) 0s P(A) S 1 for any event A; (iii) For any sequence of mutually exclusive events A1, A2 ,A", i.e. A, n Aj =...
Question 9: Let S be a set consisting of 19 two-digit integers. Thus, each element of...
Question 9: Let S be a set consisting of 19 two-digit integers. Thus, each element of S belongs to the set 10, 11,...,99) Use the Pigeonhole Principle to prove that this set S contains two distinct elements r and y, such that the sum of the two digits of r is equal to the sum of the two digits of y. Question 10: Let S be a set consisting of 9 people. Every person r in S has an age...
Question 4. (a) Let c be a cluster point of a set S. Prove directly from the e, o definition of c...
Question 4. (a) Let c be a cluster point of a set S. Prove directly from the e, o definition of continuity that the complex valued function f() is continuous within S at the point c if and only if both of the functions Re[f(a) and Im[f(2)] are continuous within S at the point c (b) For which complex values of (if any) do the following sequences converge as n → oo (give the limits when they do) and for...
JUST DO QUESTION 4 Université d'Ottawa Faculté de génie University of Ottawa Faculty of Engineeing École...
JUST DO QUESTION 4
Université d'Ottawa Faculté de génie University of Ottawa Faculty of Engineeing École de science informatique et de génle électrique uOttawa School of Electrical Engineering and Computer Science Canada's universiry ELG 3126 RANDOM SIGNALS AND SYSTEMS Winter 2018 ASSIGNMENT 1 Set Theory (due at 11.30 AM Thusday, Jan. 18 in class) I. Your University of Ottaa stdent number has k distinct digits in it. State the set of t and all the subsets of this set that...
need help with discrete math HW, please try write clearly and i will give a thumb up thanks!! (i) Prove that every comp...
need help with discrete math HW, please try write clearly and i
will give a thumb up thanks!!
(i) Prove that every complete lattice has a unique maximal element. (ii) Give an example of an infinite chain complete poset with no unique maximal 1 element (iii) Prove that any closed interval on R ([a, b) with the usual order (<) is a complete lattice (you may assume the properties of R that you assume in Calculus class) (iv) Say that...
(a) x(G) 3 d +1 b) some stable set in G has size > n Does either (a) or (b) necessarily hold when (i) d is the maximum degree of G? (ii) d is the maximum, over all non-null subgraphs H of G, of th...
(a) x(G) 3 d +1 b) some stable set in G has size > n Does either (a) or (b) necessarily hold when (i) d is the maximum degree of G? (ii) d is the maximum, over all non-null subgraphs H of G, of the minimum degree of H? (iii) d is the average degree of G?
(a) x(G) 3 d +1 b) some stable set in G has size > n Does either (a) or (b) necessarily hold when...
10. (18 points total: 3 points for each correct answer, 0 points for incorrect answers or...
10. (18 points total: 3 points for each correct answer, 0 points for incorrect answers or no answer Answer "True" or "False” for each of the following: (i) If 8,9:R + R and are both continuous at a number c, then the composition function fog is continuous at c. (ii) If functions hi, h2: R + R and are both uniformly continuous on a non-empty set of real numbers E, then the product h h2 is uniformly continuous on E....
1. (a) Let d be a metric on a non-empty set X. Prove that each of...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
I need help with 1.24 EXERCISE 1.20. Prove that every subspace VCR has an orthonora basis....
I need help with 1.24
EXERCISE 1.20. Prove that every subspace VCR has an orthonora basis. HINT: Begin with an arbitrary basis. Do the following one besi member at a time: subtract from it ils projection onto the son of the premio basis members, and then scale it to make it of unit length. This is called the Grm-Schmidt process EXERCISE 1.21. Prove Lemma 1.16 EXERCISE 1.22. Is the converse of part (1) of Proposition 1.17 true? EXERCISE 1.23. Let...
12. Definition : Let Λ be a non-empty set. If for each a є Л there is a set Aa, the collection (Aa : α Ε Λ is called an indexed collection of sets. The set A is called the index set. Traditionally Λ is often the natural numbers-you are probably pretty used to seeing sets indexed by the natural numbers but it can in fact be any other set! Here's the exercise: Let Л-R+ (meaning the positive real numbers,...
1. [15 pts] Use Definition 1.5 (definition of probability function) to prove Propo- sition 1.3 () 15 pts) & (iv) [10 pts). You do not need to prove (i) and (ii). [Definition 1.5/ Let Ω be a set of all possible events. A probability function P : Ω → 0,11 satisfies the follouing three conditions (i) 0s P(A) S 1 for any event A; (iii) For any sequence of mutually exclusive events A1, A2 ,A", i.e. A, n Aj =...
Question 9: Let S be a set consisting of 19 two-digit integers. Thus, each element of S belongs to the set 10, 11,...,99) Use the Pigeonhole Principle to prove that this set S contains two distinct elements r and y, such that the sum of the two digits of r is equal to the sum of the two digits of y. Question 10: Let S be a set consisting of 9 people. Every person r in S has an age...
Question 4. (a) Let c be a cluster point of a set S. Prove directly from the e, o definition of continuity that the complex valued function f() is continuous within S at the point c if and only if both of the functions Re[f(a) and Im[f(2)] are continuous within S at the point c (b) For which complex values of (if any) do the following sequences converge as n → oo (give the limits when they do) and for...
JUST DO QUESTION 4
Université d'Ottawa Faculté de génie University of Ottawa Faculty of Engineeing École de science informatique et de génle électrique uOttawa School of Electrical Engineering and Computer Science Canada's universiry ELG 3126 RANDOM SIGNALS AND SYSTEMS Winter 2018 ASSIGNMENT 1 Set Theory (due at 11.30 AM Thusday, Jan. 18 in class) I. Your University of Ottaa stdent number has k distinct digits in it. State the set of t and all the subsets of this set that...
need help with discrete math HW, please try write clearly and i
will give a thumb up thanks!!
(i) Prove that every complete lattice has a unique maximal element. (ii) Give an example of an infinite chain complete poset with no unique maximal 1 element (iii) Prove that any closed interval on R ([a, b) with the usual order (<) is a complete lattice (you may assume the properties of R that you assume in Calculus class) (iv) Say that...
(a) x(G) 3 d +1 b) some stable set in G has size > n Does either (a) or (b) necessarily hold when (i) d is the maximum degree of G? (ii) d is the maximum, over all non-null subgraphs H of G, of the minimum degree of H? (iii) d is the average degree of G?
(a) x(G) 3 d +1 b) some stable set in G has size > n Does either (a) or (b) necessarily hold when...
10. (18 points total: 3 points for each correct answer, 0 points for incorrect answers or no answer Answer "True" or "False” for each of the following: (i) If 8,9:R + R and are both continuous at a number c, then the composition function fog is continuous at c. (ii) If functions hi, h2: R + R and are both uniformly continuous on a non-empty set of real numbers E, then the product h h2 is uniformly continuous on E....
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
I need help with 1.24
EXERCISE 1.20. Prove that every subspace VCR has an orthonora basis. HINT: Begin with an arbitrary basis. Do the following one besi member at a time: subtract from it ils projection onto the son of the premio basis members, and then scale it to make it of unit length. This is called the Grm-Schmidt process EXERCISE 1.21. Prove Lemma 1.16 EXERCISE 1.22. Is the converse of part (1) of Proposition 1.17 true? EXERCISE 1.23. Let...
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