discrete
EXERCISE
2.4.2: Symmetric difference applied to many sets.
(a)Calculate A ⊕ B ⊕ C for A = {1, 2, 3, 5}, B = {1, 2, 4, 6}, C = {1, 3, 4, 7}.
Note that the symmetric difference operation is associative: (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C).
(b)Let A, B, and C be any finite sets. Give a concise description for which elements from A, B, and C are in A ⊕ B ⊕ C.
(c)Let A1, A2, …, An be any finite sets. Give a concise description for which elements from A1, A2, …, An are in A1 ⊕ A2 ⊕ … ⊕ An.
Homework Answers
A. (Leftovers from the Proof of the Pigeonhole Principle). As before, let A and B be finite sets with A! 〉 BI 〉 0 and let f : A → B be any function Given a A. let C-A-Va) and let D-B-{ f(a)} PaRT...
A. (Leftovers from the Proof of the Pigeonhole Principle). As before, let A and B be finite sets with A! 〉 BI 〉 0 and let f : A → B be any function Given a A. let C-A-Va) and let D-B-{ f(a)} PaRT A1. Define g: C -> D by f(x)-g(x). Briefly, if g is not injective, then explain why f is not injective either. Let j : B → { 1, 2, 3, . . . , BI}...
1.9.1 Definition. The rth elementary symmetric function Ef(xi,2,.. ,.xn) is the sum of all possib...
1.9.1 Definition. The rth elementary symmetric function Ef(xi,2,.. ,.xn) is the sum of all possible products of r elements chosen from fxi,*2, .. . *n) with- out replacement where order doesn't matter. Prove these three facts about elementary symmetric functions for any ai, a2, and any nonnegative integer r. . an, (b): Er(-a1 ,-a2 ,-an)-(-1 )EXa1 ,a2, ,an).
1.9.1 Definition. The rth elementary symmetric function Ef(xi,2,.. ,.xn) is the sum of all possible products of r elements chosen from fxi,*2,...
discrete math Need 7c 9ab 10 15 16 17 (7) Consider the following matrices. Compute the...
discrete math
Need 7c 9ab 10 15 16 17
(7) Consider the following matrices. Compute the following matrices A=[ ]B=[ 1 c-[! (a) CA (b) BAA (c) AOC (9) Determine if the following statements are True or False. If the statement is False, explain why. (a) Consider A={1,2,3,4,5). Do A1 = {1,3,5}, A2 = {2,4}. (i) Show that P ={A1, A2} forms a partition of A. (ii) Construct the matrix of the relation R corresponding to P (b) Consider A...
Question 2 please Exercise 1. Define an operation on Z by a b= a - b....
Question 2 please
Exercise 1. Define an operation on Z by a b= a - b. Determine ife is associative or commutative. Find a right identity. Is there a left identity? What about inverses? Exercise 2. Write a multiplication table for the set A = {a,b,c,d,e} such that e is an identity element, the product is defined for all elements and each element has an inverse, but the product is NOT associative. Show by example that it is not associative....
Proposition 4.1: Here is the question: Proposition 4.1. Let (E,E. u) be a measure space. Then...
Proposition 4.1:
Here is the question:
Proposition 4.1. Let (E,E. u) be a measure space. Then the following holds for all measurable sets A, B and A, A2,.. (we do not require them to be disjoint): (Finite additivity): AnB = 0 (AUB) /(A)ja(B), (Monotonicity): A C B u(A) < i(B) (A) (Sequential continuity): An A >u(An) If u(A1) and A,\A, then u(An)/(A) (U An) < E1 4(An). (Boole's inequality): fa Give an example of a measure space where the second...
Solve problem 2 using the priblem 1 . Question is taken from Ring theory dealing with ideals and ...
Solve problem 2 using the priblem 1 . Question is taken from
Ring theory dealing with ideals and generating sets for
ideals.
Problem 1. Suppose that R (R,+ Jis a commutative ring with unity, and suppose F- (a,,. , a } is a finite nonempty subset of R. Modify your proof for Problem 5 above to show that 7n j-1 Problem 2. Consider the set Zo of integer sequences introduced in Homework Problem 6 of Investigation 16. You showed that...
help please and thank you 3. The symbole, also called XOR, is the logic operation modeling...
help please and thank you
3. The symbole, also called XOR, is the logic operation modeling the exclusive OR. We will define Peas -(P Q). (a) Give a truth table that fully describes P Q. (Just like the other logic operations were defined in Lecture 3.) (b) State what it would mean for 2 to be commutative and associative, and then prove or disprove your statements. (c) Let A, B be sets. Prove that r e AAB iff (x E...
Let S = {x ER:[x]<1}=(-1,1). We will refer to E as hyperbolic relativity space. Now a+b...
Let S = {x ER:[x]<1}=(-1,1). We will refer to E as hyperbolic relativity space. Now a+b define a binary operation by: if a,beR and ab +-1, then aob= 1+ ab Proposition 1. (5,0) is a group. Remark. This is the kind of problem that every student should become competent at doing. Perhaps some of the details here are more challenging than normally but understanding what are the steps to follow in such a problem is basic, and everyone should understand...
Exercise 4. [10 marks For every n EN, we define the union and intersection of a...
Exercise 4. [10 marks For every n EN, we define the union and intersection of a collection of n sets A1, A2, ..., An as n UAk = {2 : 3k € {1,...,n} Te Ak} and Ag = {2: Vk € {1,..., n} TE Ax}. k=1 k=1 We define the union and intersection of an infinite collection of sets A1, A2, ...,Ak,... as Ū4x = {2: JkEN 7€ Ax} and ņ 4x = {#: V EN 1 € Ag}. k=1...
1. What is wrong with the following proof that shows all integers are equal? (Please explain...
1. What is wrong with the following proof that shows all integers are equal? (Please explain which step in this proof is incorrect and why is it so.) Let P(n) be the proposition that all the numbers in any set of size n are equal. 1) Base case: P(1) is clearly true. 2) Now assume that P(n) is true. That is for any set of size n all the numbers are the same. Consider any set of n + 1...
A. (Leftovers from the Proof of the Pigeonhole Principle). As before, let A and B be finite sets with A! 〉 BI 〉 0 and let f : A → B be any function Given a A. let C-A-Va) and let D-B-{ f(a)} PaRT A1. Define g: C -> D by f(x)-g(x). Briefly, if g is not injective, then explain why f is not injective either. Let j : B → { 1, 2, 3, . . . , BI}...
1.9.1 Definition. The rth elementary symmetric function Ef(xi,2,.. ,.xn) is the sum of all possible products of r elements chosen from fxi,*2, .. . *n) with- out replacement where order doesn't matter. Prove these three facts about elementary symmetric functions for any ai, a2, and any nonnegative integer r. . an, (b): Er(-a1 ,-a2 ,-an)-(-1 )EXa1 ,a2, ,an).
1.9.1 Definition. The rth elementary symmetric function Ef(xi,2,.. ,.xn) is the sum of all possible products of r elements chosen from fxi,*2,...
discrete math
Need 7c 9ab 10 15 16 17
(7) Consider the following matrices. Compute the following matrices A=[ ]B=[ 1 c-[! (a) CA (b) BAA (c) AOC (9) Determine if the following statements are True or False. If the statement is False, explain why. (a) Consider A={1,2,3,4,5). Do A1 = {1,3,5}, A2 = {2,4}. (i) Show that P ={A1, A2} forms a partition of A. (ii) Construct the matrix of the relation R corresponding to P (b) Consider A...
Question 2 please
Exercise 1. Define an operation on Z by a b= a - b. Determine ife is associative or commutative. Find a right identity. Is there a left identity? What about inverses? Exercise 2. Write a multiplication table for the set A = {a,b,c,d,e} such that e is an identity element, the product is defined for all elements and each element has an inverse, but the product is NOT associative. Show by example that it is not associative....
Proposition 4.1:
Here is the question:
Proposition 4.1. Let (E,E. u) be a measure space. Then the following holds for all measurable sets A, B and A, A2,.. (we do not require them to be disjoint): (Finite additivity): AnB = 0 (AUB) /(A)ja(B), (Monotonicity): A C B u(A) < i(B) (A) (Sequential continuity): An A >u(An) If u(A1) and A,\A, then u(An)/(A) (U An) < E1 4(An). (Boole's inequality): fa Give an example of a measure space where the second...
Solve problem 2 using the priblem 1 . Question is taken from
Ring theory dealing with ideals and generating sets for
ideals.
Problem 1. Suppose that R (R,+ Jis a commutative ring with unity, and suppose F- (a,,. , a } is a finite nonempty subset of R. Modify your proof for Problem 5 above to show that 7n j-1 Problem 2. Consider the set Zo of integer sequences introduced in Homework Problem 6 of Investigation 16. You showed that...
help please and thank you
3. The symbole, also called XOR, is the logic operation modeling the exclusive OR. We will define Peas -(P Q). (a) Give a truth table that fully describes P Q. (Just like the other logic operations were defined in Lecture 3.) (b) State what it would mean for 2 to be commutative and associative, and then prove or disprove your statements. (c) Let A, B be sets. Prove that r e AAB iff (x E...
Let S = {x ER:[x]<1}=(-1,1). We will refer to E as hyperbolic relativity space. Now a+b define a binary operation by: if a,beR and ab +-1, then aob= 1+ ab Proposition 1. (5,0) is a group. Remark. This is the kind of problem that every student should become competent at doing. Perhaps some of the details here are more challenging than normally but understanding what are the steps to follow in such a problem is basic, and everyone should understand...
Exercise 4. [10 marks For every n EN, we define the union and intersection of a collection of n sets A1, A2, ..., An as n UAk = {2 : 3k € {1,...,n} Te Ak} and Ag = {2: Vk € {1,..., n} TE Ax}. k=1 k=1 We define the union and intersection of an infinite collection of sets A1, A2, ...,Ak,... as Ū4x = {2: JkEN 7€ Ax} and ņ 4x = {#: V EN 1 € Ag}. k=1...
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