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7. For any numbers a and b and an even natural number n, show that the...
5 numbers chosen randomly without replacement. "B" represents number of even numbers, this random variable has...
5 numbers chosen randomly without replacement. "B" represents number of even numbers, this random variable has this probability: x 0 1 2 3 4 5 p(B=x) 0.02693 0.15989 .33858 .31977 .13464 .02020 number of odd #s chosen would then be 5-x, if x is even #s chosen. "C" represents difference b/w # of even and # of odd chosen, --> C= 2B-5 a. probability that exactly 1 even # chosen? b. probability at most 1 even # chosen? c. prob....
Use the well-ordering principle of natural numbers to show that for any positive rational number x...
Use the well-ordering principle of natural numbers to show that for any positive rational number x ∈ Q, there exists a pair of integers a, b ∈ N such that x = a/b and the only common divisor of a and b is 1.
Problem 5. (20 pts) Let r,n N be two natural numbers with r < n. An...
Problem 5. (20 pts) Let r,n N be two natural numbers with r < n. An r x n matrix M consisting of r rows and n columns is said to be a Latin rectangle of size (r, n), if all the entries My belong to the set {1,2,3,..., n), for 1Si<T, 1Sj<T, and the same number does not appear twice in any row or in any column. By defini- tion, a Latin square is a Latin rectangle of size...
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k,...
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1...
10. A natural number n is called attainable if there exists non-negative integers a and b such th...
10. A natural number n is called attainable if there exists non-negative integers a and b such that n - 5a + 8b. Otherwise, n is called unattainable. Construct an 9 x 6 matrix whose rows are indexed by the integers between 0 and 8 and whose columns are indexed by the integers between 0 and 5 whose (x, y)-th entry equals 5x + 8y for any 0 < r < 8 and (a) Mark down all the attainable numbers...
1. Give a complete list of all numbers a for which z2 +1 > ar. 2....
1. Give a complete list of all numbers a for which z2 +1 > ar. 2. Definition: A function f is even if f(-x) = f(x) for all inputs z. A function f is odd if f(-x) = -f(x). (a) Let f be any function with domain (-0,0). i. Show that the function g(x) = f(x) + f(-x) is even. ii. Show that the function h(x) = f(0) - f(-x) is odd. iii. Show that f can be written as...
Done Edit Give complete solutions! Problems: (1) Recall, A,, is the alternating subgroup of Sm, i.e.,...
Done Edit Give complete solutions! Problems: (1) Recall, A,, is the alternating subgroup of Sm, i.e., the subgroup consisting of even permutations. Let n > 3. Show that #An= (Hint: this was discussed in class.) (2) Let k <n be natural numbers. Determine the parity (even or odd) of a cyclic permutation in (0102 ...ax) ES (Hint: the answer depends on the parity of k.) Moreet n be a natural number. Determine the parity of the permutation de S Share...
Let A be an m × n matrix, let x Rn and let 0 be the zero vector in Rm. (a) Let u, v є Rn be any two solutions of Ax 0,...
Let A be an m × n matrix, let x Rn and let 0 be the zero vector in Rm. (a) Let u, v є Rn be any two solutions of Ax 0, and let c E R. Use the properties of matrix-vector multiplication to show that u+v and cu are also solutions of Ax O. (b) Extend the result of (a) to show that the linear combination cu + dv is a solution of Ax 0 for any c,d...
(a) Suppose that f is continuous on [0, 1] and f(o) = f(1). Let n be 20. any natural number. Prov...
(a) Suppose that f is continuous on [0, 1] and f(o) = f(1). Let n be 20. any natural number. Prove that there is some number x such that f fx+1/m), as shown in Figure 16 for n 4. Hint: Consider the function g(x) = f(x)-f(x + 1/n); what would be true if g(x)ヂ0 for all x? "(b) Suppose 0 < a 1, but that a is not equal to 1/n for any natural number n. Find a function f...
8.5 Theorem. Let s andt be any two different natural numbers with s t. Then (2st. (). is a Pythagorean triple. The prec...
8.5 Theorem. Let s andt be any two different natural numbers with s t. Then (2st. (). is a Pythagorean triple. The preceding theorem lets us easily generale infinitely many Pythagorean triples, but, in fact, cvery primitive Pythagorean triple can be generated by chousing appropriale natural numbers s and and making the Pythagorean triple as described in the preceding thcorem. As a hint to the proof, we make a little observation. 8.6 Lemma. Let (a, b,e) be a primitive Pythagorean...
Problem 5. (20 pts) Let r,n N be two natural numbers with r < n. An r x n matrix M consisting of r rows and n columns is said to be a Latin rectangle of size (r, n), if all the entries My belong to the set {1,2,3,..., n), for 1Si<T, 1Sj<T, and the same number does not appear twice in any row or in any column. By defini- tion, a Latin square is a Latin rectangle of size...
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1...
10. A natural number n is called attainable if there exists non-negative integers a and b such that n - 5a + 8b. Otherwise, n is called unattainable. Construct an 9 x 6 matrix whose rows are indexed by the integers between 0 and 8 and whose columns are indexed by the integers between 0 and 5 whose (x, y)-th entry equals 5x + 8y for any 0 < r < 8 and (a) Mark down all the attainable numbers...
1. Give a complete list of all numbers a for which z2 +1 > ar. 2. Definition: A function f is even if f(-x) = f(x) for all inputs z. A function f is odd if f(-x) = -f(x). (a) Let f be any function with domain (-0,0). i. Show that the function g(x) = f(x) + f(-x) is even. ii. Show that the function h(x) = f(0) - f(-x) is odd. iii. Show that f can be written as...
Done Edit Give complete solutions! Problems: (1) Recall, A,, is the alternating subgroup of Sm, i.e., the subgroup consisting of even permutations. Let n > 3. Show that #An= (Hint: this was discussed in class.) (2) Let k <n be natural numbers. Determine the parity (even or odd) of a cyclic permutation in (0102 ...ax) ES (Hint: the answer depends on the parity of k.) Moreet n be a natural number. Determine the parity of the permutation de S Share...
Let A be an m × n matrix, let x Rn and let 0 be the zero vector in Rm. (a) Let u, v є Rn be any two solutions of Ax 0, and let c E R. Use the properties of matrix-vector multiplication to show that u+v and cu are also solutions of Ax O. (b) Extend the result of (a) to show that the linear combination cu + dv is a solution of Ax 0 for any c,d...
(a) Suppose that f is continuous on [0, 1] and f(o) = f(1). Let n be 20. any natural number. Prove that there is some number x such that f fx+1/m), as shown in Figure 16 for n 4. Hint: Consider the function g(x) = f(x)-f(x + 1/n); what would be true if g(x)ヂ0 for all x? "(b) Suppose 0 < a 1, but that a is not equal to 1/n for any natural number n. Find a function f...
8.5 Theorem. Let s andt be any two different natural numbers with s t. Then (2st. (). is a Pythagorean triple. The preceding theorem lets us easily generale infinitely many Pythagorean triples, but, in fact, cvery primitive Pythagorean triple can be generated by chousing appropriale natural numbers s and and making the Pythagorean triple as described in the preceding thcorem. As a hint to the proof, we make a little observation. 8.6 Lemma. Let (a, b,e) be a primitive Pythagorean...
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