Discrete Mathematics Answers

3)Let x=n+ε, where n is an integer and 0≤ε<1. If ⌊55x/7⌋=8n, then which of the following cannot be a value of x:

a) 14.3 

b) 37.7

c) 46.9

d) 51.6

4) Let x=n+ε, where n is an integer and 0≤ε<1. If ⌈33x/8⌉=5n, then which of the following is a valid value of x:

a) 5.8 

b) 4.7

c) 6.1

d) 2.5

Let f be a function from the set A to the set B. Let S and T be two disjoint subsets of A (i.e S∩T=∅); then which of the following cannot be true: 

a) if f is invertible, then f(S)∩f(T)=∅   

b) if f(S)∩f(T)≠∅, then f is a one-to-one function

c) if f(S)∪f(T)⊆B, then f is an onto function

d) none of the above

Express the following using language of Predicate Calculus, where it is understood that the people being discussed are in the courtroom. If any sentence is ambiguous, give all symbolic versions. (i) All judges are sober (ii) There is a dishonest lawyer. (iii) All defendants are innocent. (iv) Some plaintiffs are lawyers (v) Anybody who is honest and a defendant is innocent (vi) All defendants who are not sober are dishonest. 

Express the following using language of Predicate Calculus, where it is understood that the people being discussed are in the courtroom. If any sentence is ambiguous, give all symbolic versions. (i) All judges are sober (ii) There is a dishonest lawyer. (iii) All defendants are innocent. (iv) Some plaintiffs are lawyers (v) Anybody who is honest and a defendant is innocent (vi) All defendants who are not sober are dishonest

Let R be the relation on the set A {1,2,3,4,5,6,7} defined by the rule (a,b) elements of R if the integer product of (a,b) is divisible by 4. List the elements of R and its inverse?

Consider the following two sets A & B: 

A= {4, 8, 12, 16, … }

B = {1, 3, 5, 7, 9, … }

Let
be a function from z × z to , such that f(m,n) = (m*m)-(n*n).

. 

i) Show that every element of the set A has a preimage under the function f. 

Type/Insert your answer here! 

ii) Show that every element of the set B has a preimage under the function f. 

Type/Insert your answer here!

Prove that ((P Ꚛ Q) →¬R) ↔¬P is a tautology, a contradiction or contingency.

b. Determine whether each of these functions is a bijection from Z to Z.

f (n) = n2 + 1

For the Boolean functions F₁ and F₂ below, write down the corresponding

(i) Boolean expression in its disjunctive normal form;

(ii)Using the Karnaugh map method, write down the simple Boolean expression?                                                                     

(iii)Draw the logic circuit corresponding to simple Boolean expression

                      F₂ in (b) if possible 

Draw the Hasse Diagram representing the partial ordering 

 {(a, b) | a divides b} on {1, 2, 3, 4, 5, 6, 10, 15, 20, 30, 60}?