If A is a set, then suppose that f is a one-to-one function from A to P(A), the power set of A an...
Let P be the power set of {a, b, c}. A function f: P , the set of integers, follows: For A in P, f(A) = the number of elements in A. 1. Is f one-to-one? Explain. 2. Is f onto? Explain.
6. Suppose f is a function from a set with 3 elements to a set with 3 elements, which is not 1-1. What can you conclude MUST be true? A. The function is not onto B. The function is onto C. Such a function is not possible D. The cardinality of the two sets is different E. A and D F. None of the above
3 Suppose the power,P, from a circuit as a function of current, I s known to obey an equation of the form: P- al+bl+4 where "a" and "b" denote unknown constants. Suppose the actual values of I and P are found to be as shown in Table 1 below. 2 4 10 16 22 26 To a) Formulate a suitable cost function and find the least squares estimates of a and b using classical optimization technique Hint: After formulating the...
4. Let A be a non-empty set and f: A- A be a function. (a) Prove that f has a left inverse in FA if and only if f is injective (one-to-one) (b) Prove that, if f is injective but not surjective (which means that the set A is infinite), then f has at least two different left inverses.
Consider the three-dimensional subspace of function space defined by the span of 1, r, and a2 the first three orthogonal polynomials on -1,1. Let f(x) 21, and consider the subset G-{g(z) | 〈f,g〉 0), the set of functions orthogonal to f using the L inner product on, (This can be thought of as the plane normal to f(x) in the three-dimensional function space.) Let h(z) 2-1. Find the function g(x) є G in the plane which is closest to h(x)....
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...
Please show all the work nice and neat 3 Suppose the power, P, from a circuit as a function of current, I, is known to obey an equation of the form: P=a12 + bl + 4 where "a" and "b" denote unknown constants. Suppose the actual values of I and P are found to be as shown in Table 1 below. line 2 10 16 22 26 4 a) Formulate a suitable cost function and find the least squares estimates...
Define four sets of integers Let P {0, 1), let Q {-11, 1, 5) , and Let R and S be arbitrary nonempty subsets of Z. Define an even indicator function F F: ZP by F(x) = (x + 1) mod 2 for x e Z That is, F(x) 1 if x is even, and F(x) = 0 if x is odd. or neither? Explain. a) Is F: Q P one-to-one, onto, both, or neither? Explain. b) Is F: (Pn...
Let Ω be an open set and a E Ω with (the closed disc) D(a,p) Ω Let f є H(Q). We have proved that for any r 〈 ρ, f has a power series expansion in the open disc D(a,r) CO 0 where, for all n0,1,2 7l Here C is the positively oriented circle: z-a+pe.θ, 0-θ-2π. In particular, f has a Taylor series expansion in D(a, r): f" (a) 2-a 0 This results in two consequences (will be shown in...
Exercise 25: Let f: [0,1R be defined by x=0 fx)/n, m/n, with m, n E N and n is the minimal n such that z m/n x- m/n, with m,n E N and n is the minimal n such that x a) Show that L(f, P) = 0 for all partitions P of [0, 1]. b) Let m E N. Show that the cardinality of the set A :-{х є [0, 1] : f(x) > 1/m} is bounded by m(m...