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1. Dear students get fully solved SMU BSC IT assignments
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ASSIGNMENT
PROGRAM BSc IT
SEMESTER SECOND
SUBJECT CODE & NAME BT0069, Discrete Mathematics
CREDIT 4
BK ID B0953
MAX.MARKS 60
Q.1 If U = {a,b,c,d,e},A ={a,c,d}, B = {d,e},C = {b,c,e}
Evaluate the following:
(a) A’  (B-C)
(b)(AB)’(BC)
(c)(A-B)(B-C)
(d)(BC)’A
(e)(B-A)C’
Answer:
(a) A’  (B-C)
A’= setof those elementswhichbelongtoU butnot to A.
A’= (b,e)
(B-C) = (d)
A’  (B-C) = (b,e)(d)
(b)(AB)’(BC)
(AB) = (a,c, d, e)
(AB)’= (b)

2. 2 (i) State the principle ofinclusionand exclusion.
(ii) How many arrangements of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 contain at least one of the patterns
289, 234 or 487? 4+6 10
Answer:
I) Principle ofInclusionand Exclusion
For any twosets P and Q,we have;
i) |P ‫ﮟ‬ Q| ≤ |P| + |Q| where |P|isthe numberof elementsin P,and|Q|is the numberelementsinQ.
ii) |P ∩ Q| ≤
3 If G is a group, then
i) The identityelementofG isunique.
ii) Every elementinG has unique inverse inG.
iii)
For any a єG,we have (a-1)-1= a.
iv) For all a, b є G,we have (a.b)-1 = b-1.a-1. 4x 2.5 10
Answer: i) Let e, f be twoidentityelementsin G.Since e isthe identity,we have e.f=f.Since f is the
identity,we have e.f =e. Therefore, e= e.f = f. Hence the identityelementisunique.
ii)Let a be in G and a1, a2 are

3. 4 (i) Define validargument
(ii) Show that ~(P ^Q) followsfrom ~ P ^ ~Q. 5+5= 10
Answer: i)
Definition
Anyconclusion,whichisarrivedatbyfollowingthe rulesiscalledavalidconclusionandargumentis
calleda validargument.
ii) Assume ((PQ)) asan additional premise.Then,
5 (i) Construct a grammar for the language.
'L⁼{x/ xє{ ab} the number ofas in x isa multiple of3.
(ii)Findthe highesttype numberthat can be appliedto the followingproductions:
1. S→ A0, A → 1 І 2 І B0, B → 012.
2. S → ASB І b, A → bA І c ,
3. S → bS І bc. 5+5 10
Answer: i)
Let T = {a,b} and N = {S, A,B},
S isa startingsymbol.
The set of productions: 

4. 6 (i) Define tree withexample
(ii) Any connectedgraph with ‘n’ verticesand n -1 edgesis a tree.5+5 10
Answer: i)
Definition
A connectedgraphwithoutcircuitsiscalleda tree.
Example
Considerthe twotreesG1 = (V,E1) and G2 = (V,E2) where V = {a, b,c, d, e,f, g, h,i, j}
E1 = {{a,c}, {b,c}, {c,d}, {c, e},{e,g
ii)
We prove thistheorembyinductiononthe numberverticesn.
If n = 1, thenG containsonlyone vertex andnoedge.
So the numberof edgesinG is n -1 = 1 - 1 = 0.
Suppose the inductionhypothesisthatthe statementistrue forall treeswithlessthan „n‟vertices.Now
letus consideratree with„n‟ vertices.
Let „ek‟be anyedge inT whose endverticesare vi andvj.
Since T is a tree,byTheorem12.3.1,
Dear students get fully solved SMU BSC IT assignments
Send your semester & Specialization name to our mail id :
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