(a) If G is a group of even order, then
show that there exists an element $$a \ne e$$,
the identifier $$g$$, such that
$${a^2} = e$$

(b) Consider the set of integers $$\left\{ {1,2,3,4,6,8,12,24} \right\}$$ together with the two binary operations LCM (lowest common multiple) and GCD (greatest common divisor). Which of the following algebraic structures does this represent?
i) Group ii) ring
iii) field iv) lattice
Justify your answer

Maximum number of edges in a planar graph with $$n$$ vertices is _______ .

Which of the following is an example of a spooled device?

At a particular time of computation the value of a counting semaphore is 7. Then 20 P operations and 15 V operations were completed on this semaphore. The resulting value of the semaphore is: