Question

1. Find a counter-example, if possible, to these universally quantified statements (where the domain is integers). (a) Vx(x? > x) (b) Vx (x >Ovx<0) (c) Vx (x = 1) 2. Prove or disprove that the difference of the squares of two odd numbers is always divisible by 4. 3. A forest has 27 vertices and 18 edges. How many connected components does it have? Is it possible for such a graph to have just two leaves?

1,2, and 3 Please

Answer 1

1. (a) for all x in the set of integers, (x² $\geq$ x) implies,

x² - x $\geq$ 0 or, x(x -1) $\geq$ 0,

or, (x $\geq$ 0 & x $\geq$ 1) or (x $\leq$ 0 & x $\leq$ 1)

or, (x $\geq$ 1) or (x $\leq$ 1) and for x = 0, we have trivially, 0² $\geq$ 0

So, we have covered all of the set of integers by the above statement.

So, the statement is true universally in the mentioned domain.

(b) by the statement, for all x, (x > 0 v x < 0) implies x is a non-zero integer.

But, the domain is the set of all integers, which include 0 as well.

So, the element 0 serves as a counter-example.

(c) for all x (x = 1) does not hold in the set of integers for any element other than 1.

So, for example, the integer 2 serves as a counter-example.

2. Let, a and b be two arbitrary odd numbers.

Then, a = 2s + 1 and b = 2t + 1 for integers s & t.

So, a² - b² = (2s+1)² - (2t+1)² = 4s² + 4s + 1 - 4t² - 4t - 1 = 4(s²+t²+s+t)

So, a² - b² is a multiple of 4, hence, always divisible by 4.