December 1999
MA214 : DISCRETE MATHEMATICS

QUESTION 1 (Compulsory)

Total Marks: 30 Marks

• Question 2
• Question 3
• Question 4
• Question 5
• Cover Page

SUGGESTED SOLUTIONS
Solutions and allocated marks are indicated in green.
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For each of the following sets, determine whether 2 is an element of that set.

(i){x |x is an integer greater than 1 }

yes

(ii){x |x is the square of an integer }

no

(iii){2, {2} }

yes

(iv){ {2}, { {2} } }

no

(v){ {2}, {2, {2} } }

no

(vi){ { {2 } } }

no

The matrices A and B are given by

(i)Calculate [2 marks ]

Award one mark for correct multiplication of matrices and one for correctly transposing the matrix.

(ii)Calculate [2 marks ]

Either note that and use part (i),or

Award two marks for first method, for second method award one mark for correct transposes and one mark for correct multiplication.

Let p and q be the propositions

p :It is below freezing
q :It is snowing

Write the following propositions using p and q and logical connectives.

(i)It is below freezing and snowing.[1 mark ]

pq

(ii)It is below freezing but not snowing.[1 mark ]

p~q

(iii)It is not below freezing and it is not snowing.[1 mark ]

~p ~ q

(iv)It is either below freezing or it is snowing, and it is not snowing if it is below freezing.[1 mark ]

(p V q )(p ~ q )

(v)It is below freezing precisely when it is snowing.[1 mark ]

q p

A sequence of 10 bits is randomly generated. What is the probability that at least one of these bits is 0?

Let E be the event that at least one of the 10 bits is 0.Then ¯E is the event that all the bits are 1s.Since the sample space S is the set of all bit strings of length 10, it follows that

Use mathematical induction to prove that

whenever n is a nonnegative integer.

Let P (m )be the statement .One mark. Step 1. Prove P (0 ).

Let A be the set {1 ,2 ,3 ,4 }. Which ordered pairs are in the relation

R ={(1 ,1 ),(1 ,2 ),(1 ,3 ),(1 ,4 ),(2 ,2 ),(2 ,4 ),(3 ,3 ),(4 ,4 )}Two marks. Subtract one mark for each mistake up to a maximum of two.

How many ways are there to choose 6 items from 10 distinct items when

(i)the items in the choices are ordered and repetition is not allowed?
[2 marks ]

151,100

(ii)the items in the choices are ordered and repetition is allowed?
[2 marks ]

(iii)the items in the choices are unordered and repetition is not allowed?
[2 marks ]

210

For each part award one mark for the correct formula and one mark for the correct answer.