Discrete Mathematics Answers

Given two sets A and B, for each of the following statements, what can you conclude about the sets?

For example: consider the statement A−B=∅, this could be possible if -

Scenario - 1: if A=B then A−B=∅

Scenario - 2: Since A−B=A−(A∩B), if (A∩B) = A then A−B=∅

Scenario - 3: A=∅ in which case, no matter what B is A−B=∅.

Therefore we can conclude that if A−B=∅, then one of the above scenarios must be true. You do not need to draw exactly 3 conclusions. Try to answer with an exhaustive list of conclusions you can draw from each of the following statements:

(a) A∪B=A

(b) A∩B=A

(c) A−B=A

(d) A∩B=B∩A

(e)A−B=B−A

(a) Find the solution to a_n = a_n-1 + 2n + 3 with the initial conditions a₀= 4.

(b) Consider the recurrence a_n = a_n-1 + 2a_n-2 + 2n - 9 show that this recurrence is solved by:

i. a_n = 2 - n

ii. a_n = 2 - n + b * 2ⁿ for any real b.

Find out if the following sets are Countable, Uncountable, Finite or if it cannot be determined. Give the reasoning behind your answer for each.

(a) Subset of a countable set

(b) integers divisible by 5 but not by 7

(c) (3, 5)

(d) A - B (A is an Uncountable set and B is a Countable set)

(e) P(C) where C is a finite set

mathematical notations for The set of all even numbers.

LetA={0,2,4,6,8},B={0,1,2,3,4},andC={0,3,6,9}.Find the following:

1. A∪B∪C
2. A∩B∩C
3. (A∪B∪C)𝐶

4.(A∩B∩C)𝐶

5.(A∪B)∩C)𝐶 

Verify the validity of the argument: All lions are fierce. Some lions do not drink coffee. Hence some fierce creatures do not drink coffee.” (Lewis Carrol)

If (S,*) is a.semigroupand x € s show that (S,∆) is a semigroup if a∆b =a*x*b

Use a Venn diagram to illustrate the subset of odd integers in the set of all positive integers not exceeding 10.

R₁ = {(4,5)}

R₂ =    {(1,5), (1,6), (1.7), (1,8), (2,5), (2,6), (2,7), (2,8), (3,5), (3,6), (3,7), (3,8), (4,5), (4,6), (4,7), (4,8)}

Evaluate R₁ ◦ R₂ and R₂ ◦ R₁