Discrete Mathematics Answers

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}? 

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

All defendants who are not sober are dishonest

If 𝐴 and B are finite sets which are subsets of 𝑈. Establish a formula for 𝑛(𝐴 ∪ 𝐵) in terms of 𝑛(𝐴), 𝑛(𝐵) and 𝑛(𝐴 ∩ 𝐵). Hence or otherwise deduce a formula for a particular case where A and B are disjoint?

In the following argument, determine the validity or otherwise of the Statement: 

b)  “If you play football during a thunderstorm, you’ll get hit by lightning. 

           You didn’t get hit by lightning. Therefore, you didn’t play football in a  thunderstorm”

  Therefore, taxes are lowered.”

(i)                Write out the propositional statements in the above argument?

(ii)              State the premise(s) and conclusion in the argument?

(iii)            Using a truth table, determine the validity of the argument?    

In the following argument, determine the validity or otherwise of the Statement: 

a) If you aren’t polite, you won’t be treated with respect. You aren’t treated with respect. Therefore, you aren’t polite.    

For the following Boolean functions, give a tabular representation of their Boolean expressions:

(i)                  F x y z( , , ) =[x y z  ]  

(ii)                F x y z( , , ) =[x z y z  ] [ ]

What can you say about (i) and (ii) above?