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1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer...
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer d there exist unique integers q and r such that n = dq + r and 0 r< d. We define the mod function as follows: (, r r>n = qd+r^0<r< d) Vn,d E Z d0 Z n mod d That is, n mod d is the remainder of n after division by d (a) Translate the following statement into predicate logic:...
a. Define what it means for two logical statements to be equivalent b. If P and Q are two statements, show that the statement ( P) л (PvQ) is equivalent to the statement Q^ P c. Write the converse a...
a. Define what it means for two logical statements to be equivalent b. If P and Q are two statements, show that the statement ( P) л (PvQ) is equivalent to the statement Q^ P c. Write the converse and the contrapositive of the statement "If you earn an A in Math 52, then you understand modular arithmetic and you understand equivalence relations." Which of these d. Write the negation of the following statement in a way that changes the...
a) Let f : R → R be a function and CER. Definition 1. The lim+oe an A if for every e>0 there erists a M EN such that for all n 2 M we have lan - A<E Complete the following statement with ou...
a) Let f : R → R be a function and CER. Definition 1. The lim+oe an A if for every e>0 there erists a M EN such that for all n 2 M we have lan - A<E Complete the following statement with out using negative words (you do not have to prove it): The lim, 10 10if R).Consider the following subsets of P: (b) Let P2-(f(t)- ao at + azt | ao, a1, a2 and Notice that Y...
Im wondering how to do b). (6) We define the set of compactly supported sequences by...
Im wondering how to do b).
(6) We define the set of compactly supported sequences by qo = {(zn} : there exists some N > 0 so that Zn = 0 for all n >N). We define the set of compactly supported rational sequences by A={(za) E ao : zn E Q for all n E N). (a) Prove that A is countable (b) Prove that for 1 S p<oo the set A is dense in P. You may use...
Need help with 1,2,3 thank you. 1. Order of growth (20 points) Order the following functions...
Need help with 1,2,3 thank you.
1. Order of growth (20 points) Order the following functions according to their order of growth from the lowest to the highest. If you think that two functions are of the same order (Le f(n) E Θ(g(n))), put then in the same group. log(n!), n., log log n, logn, n log(n), n2 V, (1)!, 2", n!, 3", 21 2. Asymptotic Notation (20 points) For each pair of functions in the table below, deternme whether...
Please answer all parts. Thank you! 20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R...
Please answer all parts. Thank you!
20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R × S by (a, s) ~ (a, s') if there exists an s"e S such that s* (s,a-sa,) a. 0. Show that ~ is an equivalence relation on b. Let a/s denote the equivalence class...
CR, we typically think of t if : >0.. 1-1 if : <o'' this is the...
CR, we typically think of t if : >0.. 1-1 if : <o'' this is the natural way we might define the 'magnitude of a real number, but it is not the only way. a.) Prove that for ry ER, we have xy = 13. lyl. b.) Construct a new function : R-R UO) such that for r, y € R, we have: 1.) ||2||=0- I = 0 and ii.) ||3+ yll |||| + llyll but iii.) xyll ||||llyll. 36....
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z...
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z[i] (b) Prove that if a + r є z[i] is a unit of Zli], then Ma + bi)-1. (c) Find all of the units in Zli
5. Let Zli_ {a + bi l a,b...
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...
Problem 10.13. Recal that a polynomial p over R is an expression of the form p(x) an"+an--+..+ar +ao where each aj E R and n E N. The largest integer j such that a/ 0 is the degree of p. We d...
Problem 10.13. Recal that a polynomial p over R is an expression of the form p(x) an"+an--+..+ar +ao where each aj E R and n E N. The largest integer j such that a/ 0 is the degree of p. We define the degree of the constant polynomial p0 to be -. (A polynomial over R defines a function p : R R.) (a) Define a relation on the set of polynomials by p if and only if p(0) (0)...
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer d there exist unique integers q and r such that n = dq + r and 0 r< d. We define the mod function as follows: (, r r>n = qd+r^0<r< d) Vn,d E Z d0 Z n mod d That is, n mod d is the remainder of n after division by d (a) Translate the following statement into predicate logic:...
a. Define what it means for two logical statements to be equivalent b. If P and Q are two statements, show that the statement ( P) л (PvQ) is equivalent to the statement Q^ P c. Write the converse and the contrapositive of the statement "If you earn an A in Math 52, then you understand modular arithmetic and you understand equivalence relations." Which of these d. Write the negation of the following statement in a way that changes the...
a) Let f : R → R be a function and CER. Definition 1. The lim+oe an A if for every e>0 there erists a M EN such that for all n 2 M we have lan - A<E Complete the following statement with out using negative words (you do not have to prove it): The lim, 10 10if R).Consider the following subsets of P: (b) Let P2-(f(t)- ao at + azt | ao, a1, a2 and Notice that Y...
Im wondering how to do b).
(6) We define the set of compactly supported sequences by qo = {(zn} : there exists some N > 0 so that Zn = 0 for all n >N). We define the set of compactly supported rational sequences by A={(za) E ao : zn E Q for all n E N). (a) Prove that A is countable (b) Prove that for 1 S p<oo the set A is dense in P. You may use...
Need help with 1,2,3 thank you.
1. Order of growth (20 points) Order the following functions according to their order of growth from the lowest to the highest. If you think that two functions are of the same order (Le f(n) E Θ(g(n))), put then in the same group. log(n!), n., log log n, logn, n log(n), n2 V, (1)!, 2", n!, 3", 21 2. Asymptotic Notation (20 points) For each pair of functions in the table below, deternme whether...
Please answer all parts. Thank you!
20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R × S by (a, s) ~ (a, s') if there exists an s"e S such that s* (s,a-sa,) a. 0. Show that ~ is an equivalence relation on b. Let a/s denote the equivalence class...
CR, we typically think of t if : >0.. 1-1 if : <o'' this is the natural way we might define the 'magnitude of a real number, but it is not the only way. a.) Prove that for ry ER, we have xy = 13. lyl. b.) Construct a new function : R-R UO) such that for r, y € R, we have: 1.) ||2||=0- I = 0 and ii.) ||3+ yll |||| + llyll but iii.) xyll ||||llyll. 36....
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z[i] (b) Prove that if a + r є z[i] is a unit of Zli], then Ma + bi)-1. (c) Find all of the units in Zli
5. Let Zli_ {a + bi l 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...
Problem 10.13. Recal that a polynomial p over R is an expression of the form p(x) an"+an--+..+ar +ao where each aj E R and n E N. The largest integer j such that a/ 0 is the degree of p. We define the degree of the constant polynomial p0 to be -. (A polynomial over R defines a function p : R R.) (a) Define a relation on the set of polynomials by p if and only if p(0) (0)...
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