MCS-013
Descrete Mathematics
January 2016 Session
Q - 1
(a)
Ans (i) : p->(~q v ~r) ^ ~p ^ ~q
Precedence Rules
1. ~
2. ^
3. v and exor
4. -> and <->
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Ans (ii) : p->(r v ~q) ^ (~p v r)
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(b) Venn Diagram
Ans (i): (A
B)
(C~A)
C)Geometric Representation
Ans (i) : {2} x R here x=2
Ans(ii) : {1,2} x {2,-3} it has four points as shown in below figure
={(1,2), (1,-3), (2,2), (2,-3)}
Question 2:
(a)Write Down suitable mathematical statement that can be represented by the following symbolic properties.
Ans (i) : (
x)(
y)P
(any one of x)(all of y)
(all of x)(all of y)(any one of z)
(b) Show whether root of 15 is rational or irrational.
As usual, for contradiction, assume 15.5=p/q, where p,q are coprime integers and q is non-zero.
Thus, 15q2 = 5*3*q2 = p2
Since 5 and 3 are prime, they must divide p.
Hence we can say that 15 is irrational.
(c) explain inclusion - exclusion principle with example.
Book 2 Page 51-52.
Question 3:
(a)Make the logic circuit for the following expression.
Figure on Page 1 White Page.
(b)What is tautology? If P and Q are statements, show whether the statement (P->Q)V(Q->P) is a tautology or not.
A Compound Proposition that is true for all possible truth values of the simple propositions involved in it is called a tautology.
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The given statement is tautology.
Question 4:
(a) How many different 8 professionals committees can be formed each containing at least 2 Professors, at least 2 Technical Managers and 3 Database Experts from list of 10 Professors, 8 Technical Managers and 10 Database Experts?
The 8 members team can be either of the following:
a. 3 professor, 2 technical managers and 3 database expert
Total ways of choosing this =10C3*8C2*10C3=120*28*120=403200
b. 2 professor, 3 technical managers and 3 database expert
Total ways of choosing this =10C2*8C3*10C3=45*56*120=302400
c. 2 professor, 2 technical managers and 4 database expert
Total ways of choosing this =10C2*8C2*10C4=45*28*210=264600
Total ways
= 403200+302400+264600
= 970200
(b) What are De Morgan’s Law? Explain the use of De Morgan’s Law with example.
Ans:Book1 Page no 17 (a) to (e) law and Page no 18. Example (7) with solution.
(c) Explain the addition theorem in probability.
Book2 Pageno 29. Start to Example 2 and Solution.
Question 5:
(a) How many words can be formed using letter of UNIVERSITY using each letter at most once?
i) If each letter must be used,
ii) If some or all the letters may be omitted.
U
N
I
V
E
R
S
I is repeated.
T
Y
Only I is repeated hence 9 distinct character are there.
(i) if each letter must be used
Then 9! = 9x8x7x6x5x4x3x2x1=3,62,880
(ii) if some or all the letters may be omitted
0 of 9 letters
P(9,0) =0! =1
P(9,1) =9!/8! =9
P(9,2) =9!/7! =72
P(9,3) =9!/6! =504
P(9,4) =9!/5! =3024
P(9,5) =9!/4! =15,120
P(9,6) =9!/3! =60,480
P(9,7) =9!/2! =1,81,440
P(9,8) =9!/1! =3,62,880
P(9,9) =9! =3,62,880
Total : 9,86,409
(b) Show that (p->q)->q => pvq
Proved solution p->q = ~pvq
=~pvq ->q
=~(~pvq) v q
=(p^~q)vq [De Morgan’s Law of Double Negation]
=(pvq)^(~qvq) [De Morgan’s Law of Distribution]
=(pvq)^T [~qvq is always true]
=pvq
(c) prove that n!(n+2) = n!+(n+1)!
RHS = n! + (n+1)n!
=(1)n! + (n+1)n!
=(1+n+1)n!
=(n+2)n!
=LHS
Question 6:
(a)
How many ways are there to distribute 20 district object into 10
distinct boxes with:
i) At least three empty box.
ii) No empty box
Select 10
No of ways to select 3 distinct boxes: 10*9*8 = 720
No of ways to arrange 20 in the 8 remaining distinct boxes:
20P8 = 5079110900
Ans: 720*5079110900 = 4.11408x10^12
ii) No empty box.
20P10 = 20!/10! = 6.7044x10^11 ways
(b) Explain principle of multiplication with an example.
Book 2 Page no 28, topic 2.2 and example1 with solution.
(c) Set A,B and C are: A = {1, 2, 4, 8, 10 12,14}, B = { 1,2, 3 ,4, 5 } and C { 2, 5,7,9,11, 13}.
A
B
C = {1,2,4,5,7,9,11,13}
A
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C = {1,2,4,5,7,9,11,13}
A
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C = {1,2,3,4,5,8,10,12,14}
B~C = {1,3,5}
Question 7.
(a) Find how many 3 digit numbers are odd?
Ans:
1st Position -> 1-9 not zero -> 9 places
2nd Position -> 0-9 -> 10 places
3rd Position -> 1,3,5,7,9 -> 5 places of
Total 3 digit numbers = 9x10x5
= 450
(b) What is counterexample? Explain with an example.
A counterexample is a special kind of example that disproves a statement or proposition. Counterexamples are often used in math to prove the boundaries of possible theorems. In algebra, geometry, and other branches of mathematics, a theorem is a rule expressed by symbols or a formula.
Counterexamples are helpful because they make it easier for mathematicians to quickly show that certain conjectures, or ideas, are false. This allows mathematicians to save time and focus their efforts on ideas to produce provable theorems.
A counterexample to the statement "all prime numbers are odd numbers" is the number 2, as it is a prime number but is not an odd number. Neither of the numbers 7 or 10 is a counterexample, as neither contradicts the statement. In this example, 2 is the only possible counterexample to the statement, but only a single example is needed to contradict "All prime numbers are odd numbers". Similarly the statement "All natural numbers are either prime or composite" has the number 1 as a counterexample as 1 is neither prime nor composite.
(c) What is a function? Explain following types of functions with example i) Surgective ii) Injective iii) Bijective
Book2 Page no 16 for function definition. page no 18 1.5.1 types of functions with example
Question 8.
(a) Find inverse of the following function: f(x)=(x3+2)/(x-3) where x!=3
y=(x3+2)/(x-3) replace x by y and y by x
x=(y3+2)/(y-3)
x*(y-3)=(y3+2)
x*(y-3)-2=y3
xy-3x-2=y3
-3x-2=y3-xy
xy-y3=3x+2
y=((3x+2)/(x-1))1/3
f-1(x)=((3x+2)/(x-1))1/3
(b) Explain equivalence relation with example.
Ans:
Book2 Page No 15 Last Two paragraph and page 16 first five lines.
(c) Find Boolean expression for the output of the following logic
circuit.
Ans : (((A’^B)^B’)’VC)’
(d) Prove that the inverse of one-one onto mapping is unique.
Ans : White Page- 2nd Page
- Some Figures are missing in this Assignment.
- In Comments PUT Your Email ID to Receive the original Images who have the solution.
Thank you,
Hitesh Vataliya.
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