For each ne N, determine (with proof!) the number of subsets of the set [n] the sum of whose elements is odd. For example, for n = 2, there are 2 such subsets of [2], namely {1} (the sum of the elements is 1) and {1,2} (the sum of the elements is 3).

Q: Find the sup and inf of the following set ∩(1 − 1/n, 2 + 1/n) where n ∈ N

A: The problem is asking us to find the supremum (least upper bound) and infimum (greatest lower bound)…

Q: How many k-lists {r1, x2, ..., xk} are possible, such that each x; is a positive integer and 1 <x1 <…

A: Here each xi is a positive integer and 1≤x1≤x2≤...≤xk≤n.

Q: 8) Define a set X recursively as follows: B) 30 e X R,) If xe X and x is even, then e X 2 R,) If xe…

A:

Q: If n + 1 integers are chosen from the set (1, 2, 3,..., 2n). where n is a positive integer, must at…

A:

Q: (c) Prove that if A and B are finite sets, then A B is finite and A B| = |B|Al. (Hint: Use induction…

A: For (c), We are given that if A and B are finite sets. So, let |A| = m and |B| = n. Now, we need to…

Q: Assume that the universal set is Z. Consider the following sentence: (3t = Z) (t •x = 20). (a)…

A: Here, in the question, it says, such that tx= 20.We know a statement in logic and mathematics is a…

Q: 4. Show that if n is a positive integer, then Σ {a₁,...,ak}{1,2,...,n} (Here the sum is over all…

A: Given, If n is a positive integer, then we have to show that ∑a1,.....ak⊆1,2,....,n 1a1a2....ak=n

Q: Four actions are to be performed in a denite order. There are ve possibilities for the first action;…

A: Basic Counting Rule (BCR)If there are r actions can be done in a definite order, there are…

Q: Consider strings of length n, n ≥ 1, over the set (a, b, c, d). (a) How many such strings contain at…

A: (a)Total strings that contain at least one pair of adjacent characters that are the same = Total…

Q: Four actions are to be performed in a definite order. There are five possibilities for the first…

A: Basic Counting Rule (BCR): If k actions can be done in a definite order with m1 possibilities for…

Q: |(A ∪ B)\(C ∪ D)|

A: We have to give a formula for |(A ∪ B)\(C ∪ D)| in terms of the sizes of the four sets A, B, C, D…

Q: Prove that the following 2 sets are equal: {4k + 5 : k ∈ Z} = {4k + 1 : k ∈ Z}.

A: Two sets are equal if and only if They have the same element  We are using here if x €A. &lt;=&gt;…

Q: Show that the set {2, 5, 8, 11, 14, . . . } is infinite.   Group of answer choices: Choose answer…

A: Topic-series

Q: An organization wants to assemble a cluster of n computers using wires and hubs. Each must have…

A: Solution of the problem as follows

Q: A pizza restaurant sells pizzas in three sizes, and a customer can choose up to three toppings,…

A: The objective of this question is to find out the total number of pizza configurations possible…

Q: Suppose that you had infinitely many people, each one wearing a uniquely numbered button: 1, 2, 3,…

A: Infinitely many people and each wearing uniquely numbered butten(1,2,3.....) Now everyone flipping…

Q: 1) Finite Cardinalities A. For any two sets, is this statement true: n(AUB)=n(A)+n(B) - n(ANB B. For…

A:

Q: Prove that: there exist infinitely many 5-tuples of practical numbers of the form (n-8, n – 4, n,…

A: It is given that 5-tuples of practical number of the form (n-8, n-4, n, n+4, n+8)

Q: Consider all n! permutations of the set {1,2,3,..., n}. We say that a number i is a fixed point of a…

A: Given: pn(k) be number of permutations of 1, 2, ..., n that has exactly k fixed points.

Q: A. Prove that for all positive natural numbers n, 13 + 23 + ... + n3 = n2⋅(n+1)2/4.

A: According to our guideline we can solve only first question of multiple question. If you want…

Q: Suppose we know that each of An, n ≥ 0, is countable. Show that (a) {A0, A₁,..., An...} is a set. If…

A: (a). To show that {A0,A1,....,An,...}  is a set we need to use the Axiom of countable choice, which…