Discrete Mathematics Answers

The probability that on any given day, Susan carries her umbrella is 0.62. The probability that it rains and Susan has her umbrella is 0.56. What is the probability that it rains given that Susan had her umbrella.

LetA={1,2,3,4,5,6,7,8,9,10}. Show that if S is any subset of A with 7 elements, then there are 2 elements of S whose sum is 10.

Let S ={100, 101, 102,....999} so that |S| = 900.
a) How many numbers in S have at least one digit that is a 3 or 7? Examples... 300, 707, 103.
b) How many numbers in S have at least one digit that is a 3 and a 7? Examples.... 736 and 377

Problem 4
Determine |A ∪ B ∪ C| when |A|= 50, |B|= 500, and
|C| 5000, if (a)A ⊆ B ⊆ C; (b)A ∩ B A ∩ C B ∩ C
∅; and (c) |A ∩ B| |A ∩ C| |B ∩ C| 3 and |A ∩ B ∩ C| 1.

If set S has n elements how many subsets have an even # of elements?

If A=∅
1. Is it an equivalence relation?
2. Is it antisymmetric?

Which of the following problems can be solved by a standard greedy algorithm?
I. Finding a minimum spanning tree in an undirected graph with positive-integer edge weights
II. Finding a maximum clique in an undirected graph
III. Finding a maximum flow from a source node to a sink node in a directed graph with positive-integer edge
capacities
(A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III

For a connected, undirected graph G=( V, E ), which of the following must be true?
I. ∑v ∈V
degree(v) is even.
II. E ≥ |V|− 1
III. G has at least one vertex with degree 1.
(A) I only
(B) II only
(C) III only
(D) I and II
(E) II and III

Twenty-five different beauty products make the following claim: 13 claims for whitening the skin, 20 claims to soften the skin, 11 claims to reduce wrinkles, 8 claims to both whitening and softening the skin, 5 claims to both whitening and reducing wrinkles, and 6 claims to both softening the skin and reducing the wrinkles.

(i) Give a suitable name for the sets. Indicate the cardinality of each set. Determine and propose set operations that can be performed on the sample data above. Justify your reasons for proposing the set operations.

(ii) Create Venn diagram to represent the set operations that you derive from Question (i) above. Analyze the Venn diagram and state the conclusion that you can derive from the Venn diagram.

(iii) Analyze how many products make all three claims. Justify your answer.





(iv) Analyze how many products claim for whitening the skin but do not claim to reduce wrinkles. Justify your answer.

(v) Give another example of how sets can be applied to group objects or elements together. Support your answer by providing detailed explanation

Let A ={1,2,3}
what is R if R is the relation ⊂ Ƥ (A)

How can you say that a relation is complete?
Complete is a property of relation.
please give me an example.

Thank you!