Let the universal set be R, the set of all real numbers, and let P= (-1,0] = {x ER|-1 < x≤0} and Q= [0,1) = {x ER |0 ≤ x < 1} i. Find PUQ ii. Find PnQ iii. Find Q-P
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- Let Z be the set of all integers. An integer a has f as a factor if a = fj for some integer j. An integer is even if it has 2 as a factor. An integer a is odd if it is not even. Prove by contradiction that an odd number cannot have an even number as a factor.Let the universal set be the set ? of all real numbers and let ? = {x ∈ R |0 <x ≤ 2},? = {X ∈ R |1 ≤ x < 4}, ? = {X ∈ R |3 ≤ X < 9}. what is BcLet Z be the set of all integers. An integer a has f as a factor if a = integer j. An integer is even if it has 2 as a factor. An integer a is odd if it is not even. Prove by contradiction that an odd number cannot have an even number as a factor.
- Prove or disprove that for any x ∈ N, x(x+1)/2 ∈ N (where N = {0, 1, 2, 3, ….}Let E : y2 = x3 + 3x + 4 be an elliptic curve over F37. 1, Find all the elements of the elliptice curve group 2, Find the order of the 3, Find a primitive element of this group and call it G. 4, Compute [30]G = G ⊕ G ⊕ · · ⊕ G (addition of 30 many G’s)Let E = {0, &} and let A = {s | s = Yı&y2& • . . &yk for k > 0, each y; E 0*, and y; Y; for i + j}. Prove that A is not regular.
- GIVEN: n red points and n blue points in the plane in general position (i.e., no 3 points are on the same line) PROVE: there exists a matching (i.e., 1-1 correspondence) between red and blue points such that the segments connecting the corresponding points do not intersect. EXTRA/HINT: describe an algorithm for finding such matchingLet f (n) and g(n) be functions with domain {1, 2, 3, . . .}. Prove the following: If f(n) = O(g(n)), then g(n) = Ω(f(n)).Let the domain be the set of all positive integers. Let: P(x): " is a perfect square". Q(z): "z is a perfect cube". Determine if the following quantified statement is true or false: 3r (Q(x) ^ P(x))
- 1: Given a fixed integer B (B ≥ 2), we demonstrate that any integer N (N ≥ 0) can bewritten in a unique way in the form of the sum of p+1 terms as follows:N = a0 + a1×B + a2×B2 + … + ap×Bpwhere all ai, for 0 ≤ i ≤ p, are integer such that 0 ≤ ai ≤ B-1.The notation apap-1…a0 is called the representation of N in base B. Notice that a0 is theremainder of the Euclidean division of N by B. If Q is the quotient, a1 is the remainder of theEuclidean division of Q by B, etc.1. Write an algorithm that generates the representation of N in base B. 22. Compute the time complexity of your algorithm.Imagine a 3D plane P in your 3D scene. An infinite number of lines can lie on that plane. Consider one set of parallel lines on this 3D plane. This set will produce one vanishing point when projected on the image plane. Consider now all possible sets of parallel line on the plane P. What is the locus of all vanishing points produced by all sets of parallel lines? [Note that you do not have to right down a formula in order to solve this. You need to use a geometric argument.]Let f and g be functions from the set of integers or the set of real numbers to the set of real numbers. We say that f ( x ) is O ( g ( x ) ), read as "f ( x ) is big-oh of g ( x )", if there are constants C and k such that | f ( x ) | ≤ C | g ( x ) | whenever x > k. KINDLY SHOW YOUR SOLUTION. 7. Generating sequences of random-like numbers in a specific range. Xi+1 = aXi + c Mod m where, X, is the sequence of pseudo-random numbers m, ( > 0) the modulus a, (0, m) the multiplier c, (0, m) the increment X0, [0, m) – Initial value of sequence known as seed m, a, c, and X0 should be chosen appropriately to get a period almost equal to m For a = 1, it will be the additive congruence method. For c = 0, it will be the multiplicative congruence method