Discrete Mathematics Answers

Use a Karnaugh map to simplify the following Boolean expression:

(x'"\\land" y'"\\land" z') v (x'"\\land" y'"\\land" z') v (x'"\\land" y'"\\land" z') v (x'"\\land" y'"\\land" z') v (x'"\\land" y'"\\land" z')

Say for each of the posets represented by the given Hasse-diagram whether the poset is

i) a lattice

ii) a complemented lattice

iii) a Boolean algebra

Give reasons for your answers

Consider the following relations on the set A={1,2,3,4,5}.

a) R₁={(1,1), (1,2), (3,3), (3,2)}

b) R₂={(1,1), (2,2), (3,2), (4,5), (5,2)}

c) R₃={(1,4), (2,5), (3,3), (4,2), (5,2)}

State, with reasons, for a), b), c) whether the relation is:

I) a function from A to A

ii) an onto function

iii) a one-to-one function

iv) an everywhere defined function

Let f:R"\\to" R be f(x)= x/1+|x|

a) Is f everywhere defined? If not, give the domain.

b) Is f onto? If not give the range.

c) Is f one-to-one? Explain.

d) Is f invertible? If so, what is f-^1?

Let f: A"\\to" B and g: B"\\to" C be functions. Show that if g o f is onto, then g is onto.

Write smallest cutset of K_3,5

Show that whether x5 + 10x3 + x + 1 is O(x4) or not?

Solve recurring relation :a_n+2 -10a_n+1+25a_n=5^n(n≥0)

If a simple graph G has p vertices and any two distinct vertices u and v of G have the

property that degGu + degGv ≥ p-1 then prove that G is connected

(a) Suppose f and g are functions whose domains are subsets of z+, the set of positive integers. Give the definition of 'f is "\\Omicron"(g)'

(b) Use the definition of 'f is "\\Omicron"(g)' to show that:

(I) 2ⁿ+27 is "\\Omicron"(3ⁿ)

(ii) 5ⁿ is not "\\Omicron"(4ⁿ)