Discrete Mathematics Answers

Let A be an 8*8 boolean matrix (i.e.,every entry is 0 or 1).if the sum of the entries in A is 51, prove that there is a row i and a column j in A such that the entries in row i and in column j add up to more than 13. Further show that there are at least 4 such pair of rows and columns.

Define
(a) Graph; (b) Null graph; (c) Isolated vertex; (d) Pendant vertex;
(e) Pseudo-graph; (f) Directed graph; (g) Adjacent nodes; (h) Incident edges;

Draw a Venn diagram of sets B,A and C where A . ⊆ ,B A ∩ C ≠ φ, B∩ C = φ
What is the universal set you have chosen? Justify your choice of sets in the
diagram.

(p->(q^r))^(not p->not q^not r))

Let A = {1; 2; 3; 4; 5; 6; 7; 8; 9; 10} and B = {1; 2; 3; 4}. Let R be the relation on P (A)
defined by:
For any X,Y element in P(A), XRY if and only if X-B = Y-B.
How many equivalence classes are there? Explain

Using the Karnaugh map method obtain the minimal sum of the products and product of sums expressions for the function
F (A, B, C, D) = Σ (1, 5, 6, 7, 11, 12, 13, 15).

Minimize the following problems using the Karnaugh maps method.
Z = f (A, B, C) = + B + AB + AC

GreenRecycle.com is a recycle firm whose main business is recycling paper. There is a particular standard to grade the recycled papers in the firm, which grades the papers as P11, P12, P13, P21, P22, P31 and P32.

a) Model the set of the papers
b) Three warehouses are used to store the papers and special accounts are assigned to the warehouses as given by:
warehouse = { X, Y, Z }
Account_warehouse = { (123,X), (456,Y), (789,Z) }

If the papers being stored in the warehouse are given as:
paper_warehouseAccount = { (P11,123), (P12,123), (P13,456), (P21,789), (P22,789), (P31,789), (P32,123) }

Find the papers stored in warehouse A.

(A \ B )∪ C = A \ (B∩ C )for any three sets A,B and C
Is the statement true or false?
Give justification in support of your answer.

Using the Karnaugh map method obtain the minimal sum of the products and product of sums expressions for the function
F (A, B, C, D) = Σ (1, 5, 6, 7, 11, 12, 13, 15).