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...
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....
Question 2. a) The zero transformation. We define the zero transformation, To: FN → Fm by To(x) = 0 VxEFN. (i) What is R(To)? (ii) Is To onto? (iii) What is N(To)? (iv) Is To one-to-one? (v) What is (To]s? b) The identity transformation. We define the identity transformation, Tj: Fn + En by Ty(x) = x V xEFN. (i) What is R(Ti)? (ii) Is T, onto? (iii) What is N(T)? (iv) Is T one-to-one? (v) What is Ti]s? Question...
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...
1. Prove or disprove the following statements: a. For any matriz A e Rmxn with Rank(A) = r, A and AT have the same fet of singular values. b. For any matrix A € Rnxn, the set of singular values is the set of eigenvalues. 2. Determine the definiteness of the quadratic form (11,12)= 9zł - 40102 + 6z. 3. A hospital trauma unit has determined that 25/6 of its patients are ambulatory and 75% are bedridden at the time...
I need help on b-e. THANK YOU
blem 3. Consider the following statement: 1 For all n EN, 12 +22 +32 + ... +n? n(n+1)(2n +1) (a) Prove the statement () using mathematical induction. We use the term closed form expression to describe an algebraic expression that involves only a fixed amount of operations (i.e. that doesn't involving adding n terms). So for example, in the proposition above, the sum of n consecutive natural numbers (12 +22 + ... +...
write a program in C++ Suppose that the alphabets are given the following values: a=1, b=2, c=3, ..., z=26. Using this rule, we define the value of a word as the sum of the values of its letters. Thus, the value of the word cat is c+a+t = 3+1+20 = 24. Furthermore, the value of the phrase "a cat" is a+c+a+t=1+3+1+20=25. The value of a sentence, paragraph, and any larger unit of writing is similarly defined. When we compute such...