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## Here's the Solution to this Question

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 $AB^2 +AC^2 = BC^2$ .