MATH1007 : Discrete Mathematics - Functions Counting and Boolean Algebra - Mathematics Assignment Help

Assignment Task :

PART A (Logic, Sets & Relations, Functions, Counting, Boolean algebra) (Start questions on a separate page.) 

a) Rewrite the sentence ‘I will enrol in MATH2907 if I do not fail MATH1007.’ into the form ‘if p then q ’ for appropriate propositions p and q . 

In addition, write down the contrapositive of the statement, and briefly comment on how the truth value of the contrapositive relates to that of the original statement. 

b) Let Q(x, y) be the statement x + y = 1 on the domain of integers. What are the truth values for the following? 

(i) Q(2, −3) (ii) ∀y ∃x Q(x, y)x y z F(x, y, z) 1 1 1 0 

1 1 0 1 

1 0 1 1 

1 0 0 1 

c) For the table at right, of a Boolean function of variables x, y and z , write down a formula for F(x, y, z) in disjunctive normal form (i.e., as a ‘sum of products’). Then minimise the resulting expression. 

0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 

2.a) Let the universal set U be the set of all koalas; let F , J and D denote respectively the subsets of females, juveniles and those koalas having a certain fungal disease. Express each of the following sets in terms of U , F , J and D. 

(i) Male juvenile koalas. 

(ii) Adult female koalas without the disease. 

b) Let R be the relation where x R y if and only if x − y is divisible by 3 where the universal set is the set of integers. 

(i) Is 42 R 0? Is 0 R 42? Why? 

(ii) Is R reflexive? State your reasons. 

(iii) Is R an equivalence relation? Why or why not? 

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