MCS 013 Discrete Mathematics

                                   MCS-013
                          Discrete Mathematics
1. (a) What is proposition? Explain whether, x-y >5 is a proposition or not.

Definition
	A proposition is that part of the meaning of a clause or sentence that is constant, despite changes in such things as the voice or illocutionary force of the clause.
	A proposition may be related to other units of its kind through interpropositional relations, such as temporal relations and logical relations.

Discussion
	The meaning of the term proposition is extended by some analysts to include the meaning content of units within the clause.
	Example: The tall, stately building fell is said to express propositions corresponding to the following: "The building is tall." "The building is stately." "The building fell."

Examples (English)

·  The common content of each of the the following utterances is a proposition: Alec ate the banana. The banana was eaten by Alec. Did Alec eat the banana? Alec, eat the banana. All these utterances may be analyzed as consisting of a predicate naming an event or state and one or more arguments naming referents that participate in that event or state. The activity is eat. The agent is Alec. The patient is a banana.

SOLUTION

Yes,  x-y >5 is a proposition

Because                 x-y > 5

Always  true for  x={6,7,8,9,10,……………………} or y ={0,1,2,3,4,…………….. }

And 

Always false for  {  x < y  , x = y or  y+5=x }

 









(b) Make truth table for followings.

(i) p→ (~q ⋁ ~ r) ⋀ (~p ⋁ r)

p	q	r	~p or ~q	p -> (~p or ~q)	~p or r	p→ (~q ⋁ ~ r) ⋀ (~p ⋁ r)
FALSE	FALSE	FALSE	TRUE	FALSE	TRUE	FALSE
FALSE	FALSE	TRUE	TRUE	FALSE	TRUE	FALSE
FALSE	TRUE	FALSE	TRUE	FALSE	TRUE	FALSE
FALSE	TRUE	TRUE	TRUE	FALSE	TRUE	FALSE
TRUE	FALSE	FALSE	TRUE	TRUE	FALSE	FALSE
TRUE	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE
TRUE	TRUE	FALSE	FALSE	TRUE	FALSE	FALSE
TRUE	TRUE	TRUE	FALSE	TRUE	TRUE	TRUE

(ii) p→ (r ⋁ q) ⋀ (~p ⋀ ~q)

p	q	r	r or q	p -> (r or q)	(~p and ~q)	p→ (r ⋁ q) ⋀ (~p ⋀ ~q)
FALSE	FALSE	FALSE	FALSE	TRUE	TRUE	TRUE
FALSE	FALSE	TRUE	TRUE	FALSE	TRUE	FALSE
FALSE	TRUE	FALSE	TRUE	FALSE	FALSE	FALSE
FALSE	TRUE	TRUE	TRUE	FALSE	FALSE	FALSE
TRUE	FALSE	FALSE	FALSE	TRUE	FALSE	FALSE
TRUE	FALSE	TRUE	TRUE	TRUE	FALSE	FALSE
TRUE	TRUE	FALSE	TRUE	TRUE	FALSE	FALSE
TRUE	TRUE	TRUE	TRUE	TRUE	FALSE	FALSE

(c) Draw a Venn diagram to represent followings:


 
 (i)  (A U B)  U ( B   ∩  C )


(ii)  (A U B)   ∩   ( C   ~  A )




(d) Give geometric representation for followings:


(i) R x { 4}; where R is a natural number





(ii) {2, 2) x ( 2, -4)




{(2,2),(2,2),(2,-4),(2,-4)}