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Solved Assignment MCS-013 Discrete Mathematics Q5

Q5.

(a). Draw a Venn diagram to represent followings:

i) (A B) U (C~A)

ii) (A B) U (B ~ C) (4 Marks)

=============Solution Start here======================

Draw a Vann diagram to represent following:

=============Solution End here======================

(b). Give geometric representation for the following:

i) { 3, 1} x R

ii) {-1, -1) x ( -2, 3) (4 Marks)

=============Solution Start here====================

Geometric representation: (i) {3,1} x R => {3} x R and {1} x R

The geometric diagram for {3} x R will be the line parallel to Y axis, x=3.

And similarly x=1, parallel to y axis.

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(c). What is principle of strong mathematical induction? In which situation this principle is used. (2 Marks)

======================Solution Start here=============

Principle of Strong Mathematical Induction:

Let P(n) be a predicate that involves a natural number n. Suppose we can show that

(i) p(m) is true for some m belong N

(ii) whenever p(m) ,p(m+1),.........,p(k) are true,

That p(k+1) is true, where k>=m we can conclude that p(n) is true for all natural number n>=m.

In the induction step we are making more assumption. That p(n) is true for every n lying between m and k , not just that p(k) is true.

To use the strong form of the PMI, we take m=1. We have seen that p(1) is true We also need to see if p(2) is true.

Then next step,

For an arbitrary k>=2 , We assume that p(n) is true for every n such that,

=================Solution End Here===================

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  Posted on Wednesday, April 22nd, 2009 at 3:25 PM under   Year 2009 | RSS 2.0 Feed



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