Prove that if x and y are real numbers, then max(x, y) + min(x, y) = x+y. [Hint: Use proof by cases with the two cases corresponding to x > y and x < u. respectively.]
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- Ql: The Collatz conjecture function is defined for a positive integer m as follows. (COO1) g(m) = 3m+1 if m is odd = m/2 if m is even =1 if m=1 The repeated application of the Collatz conjecture function, as follows: g(n), g(g(n)), g(g(g(n))), ... e.g. If m=17, the sequence is 1. g(17) = 52 2. g(52) = 26 3. g(26) = 13 4. g(13) = 40 5. g(40) = 20 6. g(20) = 10 7. g(10) = 5 8. g(5) = 16 9. g(16) = 8 10. g(8) = 4 11. g(4) = 2 12. g(2) = 1 Thus if m=17, apply the function 12 times in order to reach m=1. Use Recursive Function.38. If p is true, q is false, r is true, and s is false, what are the truth values of ~(q ⇔ s) and ~r ⇔ ~q, respectively?7. Let A and B be integers from one to five. For the following expression: a + b = 3, prove if the expression is reflexive, symmetric, asymmetric, and/or transitive
- Suppose for an international football tournament, you wish to find the probability of the country who can win the final game. You can find the odds for winning from your local bookie [A bookie sets odds, accepts and places bets, and pays out winnings on behalf of other people]. Odds are mathematically treated as a fraction. If the odds for the Morocco vs Argentina game are 5 to 1, the fraction is 1 / 5. So to convert the fraction into probability of winning, take the numerator and divide it by the denominator plus 1. Use the Frac.class in Java to help you convert any given odds into probability. Display your result as a percentage on you console.Given F (x, y, z) = y' + xyz': a. Express F as a Sum of Minterms b. Express F as a Product of Maxterms c. Express F' as a Sum of Minterms d. Express F' as a Product of Maxterms3. Given that x is an integer, what does if(x % 2 == 0) check for?
- 6 For the following pairs of functions, first decide whether f(n) dominates g(n), or g(n) dominates f(n); then decide whether f = 0(g), or f = N(g), or f = 0(g), and briefly explains. i. f(n) = n²,g(n) = 1000n + 30 ii. f(n) = /n, g(n) = n} iii. f(n) = 10log2 n, g(n) = log10(n³) iv. f(n) = n100, g(n) = 1.2".Question 3: Suppose a, b e Z. If a2 (b2 - 2b) is odd, then a and b are odd.3. A technician takes X hours to visit the stores in a couple of streets in one shift, and when he finishes his tour, another technician revisits the same stores again, needing other X hours to finish the second shift, and so on. For example, if the first technician started his shift at 1:00pm, he finishes it at 7:00pm and the second technician starts his shift at 7:00pm and finishes it at 1:00am, and so on. The supervisors used to make a quick meeting with all technicians twice a day at 1:00 am and 1:00 pm, so they need to finish their tours exactly 1:00. Identify the mathematical notation for the number of shifts should be made by the technicians in order to achieve this, write the name of this mathematical value, and find it for two tours, one with X=7 and another with X = 11 .
- There are a number of plants in a garden. Each of the plants has been treated with some amount of pesticide. After each day, if any plant has more pesticide than the plant on its left, being weaker than the left one, it dies. You are given the initial values of the pesticide in each of the plants. Determine the number of days after which no plant dies, i.e. the time after which there is no plant with more pesticide content than the plant to its left. Example // pesticide levels Use a -indexed array. On day , plants and die leaving . On day , plant in dies leaving . There is no plant with a higher concentration of pesticide than the one to its left, so plants stop dying after day . Input Format The first line contains an integer , the size of the array .The next line contains space-separated integers . Constraints Sample Input 7 6 5 8 4 7 10 9 Sample Output 2 Explanation Initially all plants are alive. Plants = {(6,1), (5,2), (8,3), (4,4), (7,5),…For tax purposes an item may be depreciated over a period of several years, n.With the straight line method of depreciation, each year the item depreciates by 1/nth of itsoriginal value. With the double declining balance method of depreciation, each year the itemdepreciates by 2/nths of its value at the beginning of that year. (In the final year it is depreciatedby its value at the beginning of the year.)Write a program using **Visual Basic** that performs the following tasks:1.1 Request a depreciation of the item, the year of purchase, the cost of the item, the number ofyears to be depreciated (estimated life), and the method of depreciation. The method ofdepreciation should be chosen by clicking on one of two buttons.1.2 Display a year by year description of the depreciation.Pls Use Python Explain the Wronskian determinant test. Using the Wronskian determinant test, write the program using NumPy to determine whether the functions f(x)=e^(- 3x), g(x)=cos2x and h(x)=sin2x are linearly independent in the range (-∞, + ∞). Ps: Please also explain step by step with " # "