a$)$ Use Dijkstra's algorithm to compute the shortest distance from $a$ to all the other vertices.

$[10$ marks$]$

Using the graph below

b$)$ Draw the adjacency list and perform BFS and DFS using $A$ as a source as well as the resulting BFS tree?

$[6$ marks$]$

c$)$ Determine the in $-$ degree, out $-$ degree and hence the degree of each vertex, write out the degree sequence and derive the Handshaking Principle?

$[4$ marks$]$

In the following argument, determine the validity or otherwise of the Statement:

a$),P_1$:$(QvvR)P,P_2$:notQ,$P_3$:$P,C$:$R$

$[5$ marks$]$

b$)$ Prove by induction that

$P(n)$:$4+8+12+$dots$+4n=2n(n+1),$AAn$1$

$[5$ marks$]$

$5.$ a$)$ If $A,B$ and $C$ are three sets, complete the following De$-$Morgan's laws:

$($i$),(A^1{B}^1{)}^1=$

$($ii$),(A^1{B}^1{)}^1=$

$($iii$)A-(B^{1}C)=$

$($iv$)A-(B{C}^1)=$

$[4$ marks$]$

b$)$ Draw the Hasse Diagram representing the partial ordering divides $b$ on $\{2,3,5,6,10,11,15,25\}?$

$[6$ marks$]$

$6.$ a$)$ Let $f$:$RR$ defined by $f(x)=4x-5.$ Show that $f$ is bijective and find its inverse?

$[6$ marks$]$

b$)$ Ngyyk jogmxgs oy g mxgvnoigr xktjkxotm ul g vgxzogrre uxjkxkj ykz joyvrgekj bog znk xkrgzout ul znk vgxzogrre uxjkxkj ykz cozn gt osvrokj avcgxj uxoktzgzout. G vuotz oy jxgct lux kgin krksktz ul znk vuykz, gtj rotk ykmsktzy gxk jxgct hkzckkt znkyk vuotzy giiuxjotm zu znk xarky:."

Decrypt using and additive shift of $6$

$[4$ marks$]$

a$)$ the theorem; even.$\text{'}"?$

$\backslash$table$[[$INPUTS$,$OUTPUTS$],[x,y,z,F_1(x,y,z),F_2(x,y,z)$