Question 4

Write down the negation of the following statement without using the negation symbol

or the logical implication symbol →

3x € RVY ER ((x + y< 0) → (‡z € Nx ≤yz))

(a)

(b)

(c)

Give a counterexample to show that the following statement is false:

€ €€€ Zz € N y−² < x) → (x > 1))

(10 marks)

(Vy € Z (x + 3y)) → (x² > 224)

(3 marks)

(3 marks)

Let D = {x EZ: -10 ≤ x ≤ 20}. If the variable x has domain D, determine the

truth set of the predicate

(4 marks)/n(d)

Let D be the set of all students. Define the predicates

P(y): Test y contains very difficult questions

Q(x, y): Student x passes test y

R(x, y): Teacher x sets questions for test y

Use the predicates P(y), Q(x, y), R(x, y) to write down statements expressing the

meaning of each of the following English sentences as closely as possible:

(i)

Term-2 test does not contain very difficult questions only if Teacher Andrew

did not set any questions for the test, nor did Student Mark fail the test.

(ii)

Student Irene will pass Year-End test if Teacher Andrew had not set any

questions for the test, or the test contains only easy questions.

(iii)

There are students who fail the Year-End test if and only if the test has some

very difficult questions.

(6 marks)