Question

Let the universe of discourse be the set of all integers. Let p; q; r; s, and t be as follows: p(x):x>0,q(x):xiseven,r(x):xisaperfectsquare,s(x):xis(exactly)divisibleby4, t(x):x is (exactly) divisible by 5. (8 marks)

Write the following statements using quantifiers and logical connectives

i. At least one integer is even.

ii. There exists a positive integer that is even.

iii. If x is even, then x is not divisible by 5.

iv. There exists an even integer divisible by 5.

Answer 1

i. At least one integer is even.

$\exist(x\in Z)q(x)$

ii) There exists a positive integer that is even

$\exist(x\in Z)(p(x)\land q(x))$

iii. If x is even, then x is not divisible by 5.

$\forall(x\in Z)(q(x)\rarr\overline{f(x)})$

iv. There exists an even integer divisible by 5.

$\exists(x\in Z)(q(x)\land f(x))$