Q 1. Let S = {0, 1} × {a, b, a} × {0, 1, 2}.
(a) Write out the set S in full, as a set of triples. Be careful to write your answer using correct set notation.
(b) What is the cardinality of {a, b, a}? What is the cardinality of S?
Please explain as detailed as possible, thank you! 1. Let S={0, 1, 2, 3, . . . , 150). and let A={x E S | x+100 E S} Write the roster notation of the set A. Also, find the cardinality of the set A. 2. For each natural number n, let An be the interval An (0,2/n) and let Bn be the interval Determine the following: (b) Un1Bn 3. Let the universal set be S = {1, 2, 3, 4,...
2. Let A = ZX Q and B = R? (1) Rewrite the definition of set A and B using set builder notation. (2) Show ACB (assuming Z S R and Q SR are already given).
A2 Let S := {k1, ..., kc,} be a set of containing certain possibly equal complex numbers, and let T be the set of integers lying between (and perhaps equal to) C1 and C2. Let C be the Cartesian product of S and T. a. Write C using set-roster notation. b. Write C using set-builder notation. c. What are the possible sizes (cardinalities) of C? d. If C has a subset of cardinality 5, what conditions on S are there?...
Let S be the universal set, where: S = {1, 2, 3, ..., 18, 19, 20} Let sets A and B be subsets of S, where: Set A = {2,5, 6, 7, 8, 14, 18} Set B = {1, 2, 3, 4, 7, 9, 10, 11, 12, 14, 18, 19, 20} Find the following: The cardinality of the set (A U B): n(AUB) = The cardinality of the set (A n B): n(An B) is You may want to draw...
R2 be a random variable with E(X) u = (1, #2)T, let Q [0, 2] be 2. (10 marks) Let X a г row vector and let Ги Гі2 T21 I22. E((X-)(X -)T) r (a) Compute E(QX) and write your answer in terms of the elements of u and (5 marks) Г. (b) Compute the variance of QX and write your answer in terms of the elements of u and Г. (5 marks) R2 be a random variable with E(X)...
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 n > 1, and let S = {1, 2, 3}" (the cartesian product of {1,2,3} n times). (a) What is Sl? Give a brief explanation. (b) For 0 <k <n, let T be the set of all elements of S with exactly k occurrences of 3's. Determine |Tx I, and prove it using a bijection. In your solution, you need to define a set Ax that involves subsets and/or cartesian products with known cardinalities. Then clearly define your bijection...
Let S be a finite set with cardinality n>0. a. Prove, by constructing a bijection, that the number of subsets of S of size k is equal to the number of subsets of size n- k. Be sure to prove that vour mapping is both injective and surjective. b. Prove, by constructing a bijection, that the number of odd-cardinality subsets of S is equal to the number of even-cardinality subsets of S. Be sure to prove that your mapping is...
(a) Prove directly that the cardinality of the closed interval [0, 1] is equal to the cardinality of the open interval (0, 1) by constructing a function f : [0, 1] → (0, 1) that is one-to-one and onto. (b) More generally, show that if S is an infinite set and {a,b} C S, then [S] = |S \ {a,b}\. (The notation S \ {a,b} is used to denote the set of all s in S such that s is...
2. Let set S = {(1,0, 2), (2, 1, 0) and (0,3,3)}. S is a basis for Rs. Using the Gram-Schmidt orthonormalization process to set S, obtain an orthonormal basis B' for R. 3. Find a third order Fourier approximation for the function f(x) = T-X 2 on the interval [0, 21).