2. a) Consider the propositions ‘2+3 = 5’ and ’The Sun rises in the West’. i) Write the disjunction of the statements and give its truth value. ii) Write the conjunction of the statements and give its truth value. iii) Write the exclusive disjunction of the statements and give its truth value. (3) b) Prove or disprove the following statement. (2) “If m divides a n −b n , then m divides abn −ban also.” c) Show that any tree with exactly two vertices of degree 1 is a path. (3) d) Write down the converse of each of the following statements: (2) i) If p is a prime number and a and b are any two natural numbers and if p divides a or b, then p divides ab. ii) In a triangle 4ABC, if AB2 +AC2 = BC2 , then ∠BAC = 90◦ .

a)

i) ‘2+3 = 5 or the Sun rises in the West’

$p$: ‘2+3 = 5’ (true)

$q$: ’The Sun rises in the West’ (false)

$p\lor q$

truth value: true

ii) ‘2+3 = 5 and the Sun rises in the West’

$p\land q$

truth value: false

iii) 'Either 2+3 = 5 or the Sun rises in the West'

$p\oplus q$

truth value: true

b) false

For example: $a=3,b=2,n=2$

then: $3^2-2^2=5$ is divided by 5

$ab^n-ba^n=3\cdot2^2-2\cdot3^2=-6$ is not divided by 5

c) A vertex of degree 1 is called a leaf.

Let $A$ be a tree with exactly 2 leaves $u,v\isin V(A)$. If $A$ is not a path, it means that exists $v_i\isin V(A)$ which $d(v_i)\geq3$ (degree) and a vertex $w\isin V(A)$ where the edge

$(v_i,w)\isin E(A)$.

If $d(w)=1$ then $w$ is a leaf. That's a contradiction because $A$ only has two leaves.

If $d(w)\geq2$ then exists a vertex $y\in V(A)$ and $d(y)=1$ so that exists a $wy$-path, then $y$

is a leaf and that's a contradiction.

d)

i) If p divides ab, then p is a prime number, a and b are any two natural numbers, and p

divides a or b.

ii) In a triangle $\Delta ABC$ , if ∠BAC = 90$\degree$, then $AB^2 +AC^2 = BC^2$ .