Discrete Mathematics Answers

Find the solution of the recurrence relation: xn=3xn-1 + 1, where,  x0=4

1. How many 3-digit numbers greater than 300 can be formed using the digits 

1, 2, 3, 4, and 5, when (a) repetition is not allowed and (b) repetition is  allowed. Briefly explain the calculation process.

A∩(B−C)

 If the statement ∧ r is true, determine all combinations of truth values for p and s such that the statement 

(q→[¬p∨s])∧[¬s→r] is true.

Let A, B, and C be sets. Show that (B − A) ∪ (C − A) = (B ∪ C) – A by using Venn diagram?                                                   

Using mathematical induction prove the following:

Σn k=1  k+4/k(k + 1)(k+2) = n(3n + 7)/2(n+1)(n+2) 

. The complete m-partite graph 𝑲𝒏𝟏,𝒏𝟐,……,𝒏𝒎 has vertices partitioned into m subsets of 𝒏𝟏, 𝒏𝟐, … … , 𝒏𝒎 elements each, and vertices are adjacent if and only if they are in different subsets in the partition. For example, if m=2, it is our ever-trusting friend, a complete bipartite graph.

a. Draw these graphs: i. 𝑲𝟏,𝟐,𝟑 ii. 𝑲𝟐,𝟐,𝟑 iii. 𝑲𝟒,𝟒,𝟓,𝟏

b. How many vertices and how many edges does the complete 𝑲𝒏𝟏,𝒏𝟐,……,𝒏𝒎 have?

Decide for each of the following relations whether or not it is an equivalence relation. Give full reasons if it is an equivalence relation, give the equivalence classes.

A. Let a,b E Z. Define aRb if and only if (a/b)E Z.

B. Let a and b be integers. Define aRb if and only if3|(a-b) (is the congruence modulo 3 relation)

translate to propositional logic “To get tenure as a professor, it is sufficient to be world-famous.”

Determine the truth-values of the followings

“ 1+1=4 if and only if earth is round”

“ Milk is white if and only if birds lay eggs”

“ Sky is white if and only if 1=0”

“ 33 is divisible by 5 if and only if hours has four legs”

“ x > 5 if and only if x2 > 25”