ii) How many functions from set S to set T are one-to-one? ili) How many functions...
(1 point) How many different functions f : S + T can be defined that map the domain S = {1,2,3,...,6} to the range T = {1,2,3,..., 15) such that f is NOT one-to-one? Enter your answer in the box below. Answer =
Question 1. Consider these real-valued functions of two variables: T In (x2 + y2) (a) () What is the maximal domain, D, for the functions f and g? Write D in set notation. (ii) What is the range of f and g? Is either function onto? ii) Show that f is not one-to-one. (iv) Find and sketch the level sets of g with heights: z00, 2, 04 (Note: Use set notation, and draw a single contour diagram.) (v) Without finding...
Python 1. Open up IDLE and in a multiline comment write “Functions and Sets II,” your name, and the date. 2. Use Python to write a function f(n) that finds the sum of the first n positive integers. Do not use the sum() function or lists. Write the function from scratch. Then use a for loop to print the sums for n = 10, 50, 1000. 3. Finding the Image of a function Suppose that we have a function f...
Q1) How many different 1-to-1 functions are there from a set with 6 elements to a set with 6 elements ? Q2) Use Principle Mathematical Induction to prove that for all positive integers n. 7" + 4 +1 is divisible by 6
QUESTION 10 The equality relationon any set S is: A total ordering and a function with an inverse. An equivalence relation and also function with an inverse. A function with an inverse, and an equivalence relation with as single equivalence class equal to S An equivalence relation and also a total ordering QUESTION 11 A binary operation on a set S, takes any two elements a,b E S and produces another element c e S. Examples of binary operations include...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
I need help with number 3 on my number theory
hw.
Exercise 1. Figure out how many solutions x2 = x (mod n) has for n = 5,6,7, and then compute how many solutions there are modulo 210. Exercise 2. (a) Find all solutions to x2 +8 = 0 (mod 11). (b) Using your answer to part (a) and Hensel's Lemma, find all solutions to x2 +8 = 0 (mod 121). Exercise 3. Solve f(x) = x3 – x2 +...
Problem 5. Letf: Z+Zbyn -n. Let D, E S Z denote the sets of odd and even integers, respectively. (a) Prove that fD CE, where D denotes the image of D under f. (b) Is it true that D = E? Prove or disprove. (c) Describe the set f[El. Problem 6. Letf: R R be the function defined by fx) = x2 + 2x + 1. (a) Prove that f is not injective. Find all pairs of real numbers T1,...
Number theory: Part C and Part D please!
QUADRA range's Four-Square Theorem) If n is a natural be expressed as the sum of four squares. insmber, then n cam be expressed tice Λ in 4-space is a set of the form t(x,y, z, w). M:x,y,z, w Z) matrix of nonzero determinant. The covolume re M is a 4-by-4 no is defined to be the absolute value of Det M such a lattice, of covolume V, and let S be the...
probelms 9.1
9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...