Discrete Mathematics Answers

Simplify the following expressions using laws of logic:

1. p v ~(~p --> q)
2. [(p --> q)^ ~q] --> ~p
3. [(p v q) ^ (p --> ~r) ^ r ] --> q
4. (p v ~q) ^ (p v q)

 5. ~[p --> ~(p ^ q)]

Let S be the following statement. “If it is raining, then the ground is wet.” Translate the S into symbol State in English the converse of S State in English the contrapositive of S

Simplify the following expressions using laws of logic and put what law of logic did you use or apply.

1. p v ~(~p --> q)
2. [(p --> q)^ ~q] --> ~p
3. [(p v q) ^ (p --> ~r) ^ r ] --> q
4. (p v ~q) ^ (p v q)
5. 5. ~[p --> ~(p ^ q)]

Construct the switching network of the given compound statements. Then, construct another switching network of the equivalent statement by applying the laws of logic:

1. p v (~p v q) v (p v ~q) ] ^ ~q

2. (p → q) ^ (p ↔ q)

d)Out of 300 students taking discrete mathematics ,60 take coffee, 27take cocoa,36 take tea,17 take tea only,47 take chocolate only,7 take chocolate and cocoa,3 take chocolate, tea and cocoa,20 take cocoa only ,2 take tea, coffee and Chocolate, 30 take coffee only ,9take tea and chocolate whereas 12 take tea and coffee. 

i)Express this information on a Venn diagram                                                                

 ii) Find how many Take beverages                                                                                                    

 iii) Find how many take cocoa and tea                                                                   

In a class of 200 students,70 offered physics,90 chemistry,100 mathematics while 24 did not offer any of the three subject,23 student offered physics and chemistry,41 chemistry and mathematics while 8 offered all the three subjects.How many students offered exactly two of the subjects?

how many three letter words can be formed by the word letter EQUATION

How many Combinations of bit strings length 9 have:                       a)    exactly three 0s? 

b)    at least seven 1s?

A binary operation € on three variables A, B and C define as return true if at least two true return us true then prove that OR operation on three variable is equivalent to which of the following ?

A€(B€C)        2) A€(B€¬C)   3) A€(¬B€C)     4) ¬A€(B€C)

5) A€(¬B€¬C)     6) ¬A€(B€¬C)    7) ¬A€(¬B€C)    8) ¬A€(¬B€¬C)

A town has two type of people Knave always speak lie and knative always speak truth, you meet a group of six people A, B, C, D, E and F they communicate with you as following,

A Said: All are the knative .

B Said: Atleast three are knative 

C Said: A, B are knaves 

D Said: All are Knaves except me

E Said: A, C are knaves