Question

We denote |S| the number of elements of a set S. (1) Let A and B be two finite sets. Show that if A ∩ B = ∅ then A ∪ B is finite and |A ∪ B| = |A| + |B| . Hint: Given two bijections f : A → N|A| and g : B → N|B| , you may consider for instance the function h : A ∪ B → N|A|+|B| defined as h (a) = f (a) if a ∈ A g (a) + |A| if a ∈ B. (2) Let A and B be two finite sets (not necessarily disjoint). Show that A ∪ B is finite. Hint: It may be useful to show that A∪B = A∪(B\A) and A∩(B\A) = ∅.

Answer #1

Let S be a finite set and let P(S) denote the set of all subsets
of S. Define a relation on P(S) by declaring that two subsets A and
B are related if A and B have the same number of elements.
(a) Prove that this is an equivalence relation.
b) Determine the equivalence classes.
c) Determine the number of elements in each equivalence
class.

1)Let the Universal Set, S, have 97 elements. A and B are
subsets of S. Set A contains 45 elements and Set B contains 18
elements. If Sets A and B have 1 elements in common, how many
elements are in A but not in B?
2)Let the Universal Set, S, have 178 elements. A and B are
subsets of S. Set A contains 72 elements and Set B contains 95
elements. If Sets A and B have 39 elements...

Let S denote the set of all possible finite binary strings, i.e.
strings of finite length made up of only 0s and 1s, and no other
characters. E.g., 010100100001 is a finite binary string but
100ff101 is not because it contains characters other than 0, 1.
a. Give an informal proof arguing why this set should be
countable. Even though the language of your proof can be informal,
it must clearly explain the reasons why you think the set should...

Let (X, d) be a metric space, and let U denote the set of all
uniformly continuous functions from X into R. (a) If f,g ∈ U and we
define (f + g) : X → R by (f + g)(x) = f(x) + g(x) for all x in X,
show that f+g∈U. In words,U is a vector space over R. (b)If f,g∈U
and we define (fg) : X → R by (fg)(x) = f(x)g(x) for all x in X,...

6. Let S be a finite set and let P(S) denote the set of all
subsets of S. Define a relation on P(S) by declaring that two
subsets A and B are related if A ⊆ B.
(a) Is this relation reflexive? Explain your
reasoning.
(b) Is this relation symmetric? Explain your
reasoning.
(c) Is this relation transitive? Explain your
reasoning.

Let S = {A, B, C, D, E, F, G, H, I, J} be the set consisting of
the following elements:
A = N, B = 2N , C = 2P(N) , D = [0, 1), E = ∅, F = Z × Z, G = {x
∈ N|x 2 + x < 2}, H = { 2 n 3 k |n, k ∈ N}, I = R \ Q, J =
R.
Consider the relation ∼ on S given...

Let A be a finite set and let f be a surjection from A to
itself. Show that f is an injection.
Use Theorem 1, 2 and corollary 1.
Theorem 1 : Let B be a finite set and let f be a function on B.
Then f has a right inverse. In other words, there is a function g:
A->B, where A=f[B], such that for each x in A, we have f(g(x)) =
x.
Theorem 2: A right inverse...

Let
n be a positive integer and let S be a subset of n+1 elements of
the set {1,2,3,...,2n}.Show that
(a) There exist two elements of S that are relatively prime,
and
(b) There exist two elements of S, one of which divides the
other.

We denote {0, 1}n by sequences of 0’s and 1’s of
length n. Show that it is possible to order elements of {0,
1}n so that two consecutive strings are different only
in one position

Proposition 16.4 Let S be a non–empty finite set.
(a) There is a unique n 2 N1 such that there is a 1–1
correspondence from {1, 2,...,n} to S.
We write |S| = n. Also, we write |;| = 0.
(b) If B is a set and f : B ! S is a 1–1 correspondence, then B is
finite and |B| = |S|.
(c) If T is a proper subset of S, then T is finite and |T| <...

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