Discrete Mathematics Answers

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?   

Give an indirect proof of the theorem; “If 𝑛 is an integer and 

           n³ +13is odd, then n is even.”?  

Check whether the relation R on the set S = {1, 2, 3} is an equivalent relation where:                                           

 𝑅 = {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of the following properties R has: reflexive, symmetric, anti-symmetric,  transitive? Justify your answer in each case?

Let 𝑆 = {𝑎, 𝑏, 𝑐} and 𝑅 = {(𝑎, 𝑎), (𝑏, 𝑏), (𝑐, 𝑐), (𝑏, 𝑐), (𝑐, 𝑏)}, find [𝑎], [𝑏] and [𝑐] (that is the equivalent class of a, b, and c). Hence or otherwise  find the set of equivalent class of 𝑎, 𝑏 and 𝑐?

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

verify each of the following equivalences using basic equivalences

1)((P∧Q∧R)→S∧(R→(P ∨ Q ∨ S))≡R∧(P↔Q)→S

2)((P∧Q)→R)∧(Q→(S∨R))≡Q∧(S→P)→R