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.
1. Since there are elements like "1" & "2" in Z, which are the images of more than one element of P, thus f is not one-to-one.
2. Since all the elements of Z are not images of elements of P and there are some unused elements in Z, thus f is not onto.
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 and let B-{a є A l a ¢ f( three different functions from A to P(A) and construct the set B: a)j. For the following sets, give examples of at least (b) A= {1.2.3 } 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 and...
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 Z denote the set of integers. Define function f :Z + Zby f(x) = 5; if x is even and f(x) = x if x is odd. Then f is Select one: a. One-one and onto b. Neither one-one nor onto O c. One-one but not onto O d. Onto but not one-one
1. a) Let A = {2n|n ∈ ℤ} (ie, A is the set of even numbers) and define function f: ℝ → {0,1}, where f(x) = XA(x) That is, f is the characteristic function of set A; it maps elements of the domain that are in set A (ie, those that are even integers) to 1 and all other elements of the domain to 0. By demonstrating a counter-example, show that the function f is not injective (not one-to-one). b)...
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
Question 1: Let R be the set of real numbers and let 2R be the set of all subsets of the real numbers. Prove that 2 cannot be in one-to-one correspondence with R. Proof: Suppose 2 is in one-to-one correspondence with R. Then by definition of one- to-one correspondence there is a 1-to-1 and onto function B:R 2. Therefore, for each x in R, ?(x) is a function from R to {0, 1]. Moreover, since ? is onto, for every...
11. Let the universal set be the set U = {a,b,c,d,e,f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. Find: A ∪ B 12. Let the universal set be the set U = {a,b,c,d,e,f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. Find: b. A ∩ B 13. Let the universal set be the set U = {a,b,c,d,e,f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. Find: AC...
Let A be a set with m elements and B a set of n elements, where m, n are positive integers. Find the number of one-to-one functions from A to B.
Let A be a set with m elements and B be a set with n elements in it. -When is it possible to have a k-to-1 function f such that f : A → B? -Count the number of k-to-1 functions f such that f : A → B
A 13. Let X be a p-element set and let Y be a k-element set. Prove that the number of functions f :X >Y which map X onto Y equals k!S(p, k) S#(p, k) : A 13. Let X be a p-element set and let Y be a k-element set. Prove that the number of functions f :X >Y which map X onto Y equals k!S(p, k) S#(p, k) :