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(a)Draw a Venn diagram for sets A B and C that satisfy the following
conditions:
(b)Which of the following conditional statements are true and which
are false?Justify your answers.
(i)If 1 +1 =2,then 2 +2 =5.[1 mark ]
(ii)If 1 +1 =3,then 2 +2 =4.[1 mark ]
(iii)If 1 +1 =3,then 2 +2 =5.[1 mark ]
(iv)If 1 +1 =2,then pigs can fly.[1 mark ]
(v)If 2 +2 =4,then 1 +2 =3.[1 mark ]
(i)1
+1 =2 is true,2 +2 =5 is false,so the implication is false.[1 mark
]
(ii)1 +1 =3 is false,2 +2 =4 is true,so the implication is true.[1
mark ]
(iii)1 +1 =3 is false,2 +2 =5 is false,so the implication is true.[1
mark ]
(iv)1 +1 =2 is true,pigs can fly is false,so the implication is false.
[1 mark ]
(v)2 +2 =4 is true,1 +2 =3 is true,so the implication is true.[1 mark
]
Only award marks if the correct reason is given
(c)Let A be a 3 x 4 matrix,B be a 4 x 5 matrix,and C be a 4 x 4 matrix.
Determine which of the following products are defined and find the
size of those that are defined.
(i)AB
(ii)BA
(iii)CA
(iv)AA
[5 marks ]
(i)AB is defined,size 3 ×5.One
mark for stating AB is defined, one mark for size of AB.
(ii)BA is not defined.One mark.
(iii)CA is not defined.One mark.
(iv)AA is not defined.One mark.
[5 marks ]
(d)Use mathematical induction to prove that
whenever n is a nonnegative integer.[5 marks ]
Step 1:show that statement is true for n
=1.When n =1,LHS=1,RHS=1.Two marks.
Step2: Prove that if it is true for n then it is true for n +1.
[5 marks ]
(e)Consider the three graphs A,B and C shown below.
(i)Which graph is disconnected?[1mark ]
(ii)Which graph is a directed multigraph?[1 ark ]
(iii)Which graph is a pseudograph that is connected and not directed?[1
mark ]
Justify your answers.
(i)C is disconnected
because there is no path from a to e [1mark ]
(ii)B is a directed multigraph because it is a multigraph with edges
that have a
direction indicated.[1 mark ]
(iii)A is a pseudograph as pseudographs may have any number of edges
between vertices (including loops)and it is connected and not directed.[1
mark ]
Only award marks if both the graph and the reason is correct.
(f)There are 325 computer science students at a school.How many ways
are there to pick 3 representatives from these students?[2marks ]
 .One
mark for method, one mark for correct answer. [2 marks ]
(g)Evaluate 
[2 marks ]
(h)Two fair dice are rolled.Calculate the probablity of:
(i)both dice showing 6;[1 mark ]
(ii)neither dice showing 6;[1 mark ]
(iii)at least one of the dice showing 6.[2 marks ]
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