Discrite mathematics

Question

1.

(a) Prove that if x > 3 and y > 4, then xy > 11. [3 Marks]

(b) Determine the range of f, if f : D → R, where f(x) = x [5 Marks] 2

.

I. D = ((2, 2) II. D = [[ 7, 2]

III. D = ((4, 2]

(c) Proof by contrapositive that for x, y ∈ Z, if xy is even then either x or y

is even.

2.

(a) By contradiction, prove that ∀ x, y ∈ Z, if xy is even and y is odd then x [3 Marks]

is even.

(b) Show that the set

2/3

,

1/3

,

2/9

,

1/6

· · ·

is countable.

(c) Find the values of x and y for which (x + y, 0) = (3, x x y)

Answer 1

(a)

x>3 and y>4 are given...

Multiplication property of inequalities says:

xy > 12 > 11.

(b)

for all x in D,

the range is f(x) = x

so the range is D , which is subset of R

(c)

Suppose x is odd AND y is odd

then x = 2m+1 and y=2n+1 for integers m and n

xy = (2m+1)(2n+1) = 4mn + 2m + 2n + 1

= 2(2mn+m+n) +1

by closure property of multiplicaiton and addition

over the integers, K=2mn+m+n is also an integer.

So then xy = 2K+1 for integer K

which is ODD

That is a contradiction of the GIVEN

that xy is even.

(2)

Suppose x is ODD GIVEN that xy is even and y is odd.

Then

x = 2n+1 and y = 2m+1 for integer m and n

xy = (2n+1)(2m+1)

= 4mn + 2n + 2m + 1

= 2(2mn+n+m) + 1

Again, by closure property of

multiplication and addition over

the integers, K=2mn+n+m is also an integer.

So then xy = 2K+1 is odd which is

a CONTRADICTION

(2B) Union of countable sets is countable

(2c) x+y = 3

0 = x^2y

0 = x^2 ( 3-x)

x = 0 or x=3

(0,3) and (3,0)