STUPIDSID
1(a)
Simply the set expression \[ \left (\overline{(A\cup B)\cap C \cup B} \right ) \] with justification.
6 M
1(b)
i) Use membership table to establish the set equality of :\[ \overline { (A \cap B)\cup (\overline {A}\cap C) }=\left ( A\cap \overline {B} \right )\cup \left ( \overline {A} \cap \overline {C} \right ) \]
ii) IF a - { 1,2,3,4,5,6,7}, Determine the number of subsets of A containing elements,subsets of A 1,2, and subsets of A with even number of elements.
ii) IF a - { 1,2,3,4,5,6,7}, Determine the number of subsets of A containing elements,subsets of A 1,2, and subsets of A with even number of elements.
7 M
1(c)
The sample space of an experiment is S-{a,b,c,d,e,f,g,h}. if event A= {a,b,c} and event B- {a,c,e,g}, determine \[P_{r}(A), P_{r}(B),P_{r}(A\cap B),P_{r} (A\cap B),P_{r}(A\cup B),P_{r}(\overline{A}),P_{r}(A\cap\overline {B}) \ and \ P_{r}(\overline {A}\cup B).\]
7 M
2(a)
Construct truth table for :
\[i)\ [P{\wedge}(p\rightarrow q)]\rightarrow q \\ ii) \ [P\rightarrow q)\wedge(q\rightarrow r)]\rightarrow (p\rightarrow r)\]
\[i)\ [P{\wedge}(p\rightarrow q)]\rightarrow q \\ ii) \ [P\rightarrow q)\wedge(q\rightarrow r)]\rightarrow (p\rightarrow r)\]
6 M
2(b)
simplify the switching network using the laws of logic.
7 M
2(C)
Establish the following argument by the methods of proof by contradiction.\[P\rightarrow (q\wedge r) \\ r\rightarrow s \\
\begin{matrix}
\sim (q\wedge s) \\ \overline{\ \ \ \therefore \sim P \ \ \ }
\end{matrix}\]
7 M
3(a)
Negate and simplify each of following :
i)\exists x,[p(x)\Lambda q(x)] \\ ii) \forall x,[p(x)\wedge\sim q(x)] \\ iii)\forall x,[p(x)\rightarrow q(x)] \\ iv)\exists x,[(p(x)\vee q (x)) \rightarrow r(x)\]
i)\exists x,[p(x)\Lambda q(x)] \\ ii) \forall x,[p(x)\wedge\sim q(x)] \\ iii)\forall x,[p(x)\rightarrow q(x)] \\ iv)\exists x,[(p(x)\vee q (x)) \rightarrow r(x)\]
6 M
3(b)
Find whether the following argument is valid. No engineering student of first and second semester studies logic.\[\dfrac{Anil \ is \ an \ engineering \ student \ who \ studies \ logic}{\therefore Anil \ is \ not \ in \ second \ semester }\]
7 M
3(c)
Give : i) a direct proof ii) an indirect proof and iii) proof by contradiction for the following statement.If m is an even integer
7 M
4(a)
\[if H_{1}=1,H_{2}=1+\dfrac{1}{2}+;H_{N}=1+\dfrac{1}{2}+----+1/n\] are harmonic numbers,then prove that for all \[n\varepsilon z^{1} \sum_{1=1}^{n}H_{1}=(n+1)H_{n}-n\]
6 M
4(b)
find all \[n \varepsilon z^{+} \] prove that : \[\sum_{1=1}^{n}\frac{1}{i(i+1)} = \frac{n}{n+1}\\ \sum_{i=1}^{a}i(2^{i}) = 2+(n-1)2^{n+1}/]
7 M
4(c)
\[IF A,B_{1,}B_{2}----,A_{n}\subseteq U, then prove that : \\ A_{1}\cap A_{2} \cap ----\cap A_{n}=\overline A \cup \overline A_{2}\cup ----\cup \overline A_{n} \\ ii)If A,B_{1},B_{2}----B_{n}\subseteq U \ than \ prove \ that \ A\cap B_{1} \cup B_{2} \ \cup---- \ \cup \ B_{n} )-(A \ \cap \ B_{1} )J \ (A \ \cap \ B_{2}\) \ \cup ----(A \ \cap \ B_{n}. \]
7 M
5(a)
Find the number of ways of distributing 6 objects among 4 identical containers with some containers possibly empty.
6 M
5(b)
(i) Prove that the function :\[R \times R \rightarrow R\] defined by f(a,b)=[a+b] is commutative but not associative.
ii) prove that if 30 dictionaries in a library contains a total of 61,327 pages,then at least one of the dictionary must have at least 2045 pages.
ii) prove that if 30 dictionaries in a library contains a total of 61,327 pages,then at least one of the dictionary must have at least 2045 pages.
7 M
5(c)
Let f : \[R \rightarrow R\] be defined by
\[f(\times)= \left\{\begin{matrix} 3\times- \ 5 & for \times > 0 \\ - \ 3 \ \times + \1 & for \times \leq 0 \end{matrix}\right.\]
then determine \[f_{1}\ (-1),f^{-1} (3),f^{-1}(6),(-5,5).\]
\[f(\times)= \left\{\begin{matrix} 3\times- \ 5 & for \times > 0 \\ - \ 3 \ \times + \1 & for \times \leq 0 \end{matrix}\right.\]
then determine \[f_{1}\ (-1),f^{-1} (3),f^{-1}(6),(-5,5).\]
7 M
6(a)
give a set A with A - n and a relation R on A,let M denote the relation matrix for R then prove that :
i) R is symmetric if and only if [M = M^{1}]
[R is transitive if and only if M.M = M^{2}leq M.]
i) R is symmetric if and only if [M = M^{1}]
[R is transitive if and only if M.M = M^{2}leq M.]
6 M
6(b)
I CT A - { 1,2,3,4,6,8,12} and R BE THE ORDERING ON A defined by a \[R_{b}\] if a divides b then
i) Draw the Hasse diagram of the Poset (A,R)
II) Determine the relation matrix for R
iii) Topologically sort the Poset (A, R)
i) Draw the Hasse diagram of the Poset (A,R)
II) Determine the relation matrix for R
iii) Topologically sort the Poset (A, R)
7 M
6(C)
Let A -[{1,2,3,4,5} imes {1,2,3,4,5}, and define R on A by (x_{1} y_{1})R (x_{2} y_{2}) if x_{1} + y_{1}= x_{2} - y_{2}] < br> i) Verify that R is an equivalence relation on A
ii) Determine the equivalence classes [(1,3)],[(2,4)] and [1,1)]
Determine the partition of A induced by R.
ii) Determine the equivalence classes [(1,3)],[(2,4)] and [1,1)]
Determine the partition of A induced by R.
7 M
7(a)
In a group \[s_{6}\] let
\[\alpha =\begin{pmatrix} 1 &2 &3 &4 &5 &6 \\3 &1 &4 &6 &2 &5 \end{pmatrix}\\ \beta =\begin{pmatrix} 1&2 &3 &4 &5 &6 \\4 &2 &6 &1 &3 &5 \end{pmatrix} \\ Determine \ \alpha \beta ,\ \alpha ^{3},\beta ^{4},(\alpha \beta )^{^{-1}}\]
\[\alpha =\begin{pmatrix} 1 &2 &3 &4 &5 &6 \\3 &1 &4 &6 &2 &5 \end{pmatrix}\\ \beta =\begin{pmatrix} 1&2 &3 &4 &5 &6 \\4 &2 &6 &1 &3 &5 \end{pmatrix} \\ Determine \ \alpha \beta ,\ \alpha ^{3},\beta ^{4},(\alpha \beta )^{^{-1}}\]
6 M
7(b)
Define cyclic group and prove that every subgroup of a cyclic group is cyclic.
7 M
7(c)
Define the coding function [E: Zfrac{3}{2} ightarrow Zfrac{6}{2}] by means of parity - check
[H= egin{bmatrix} 1&0 &1 &1 &0 &0 \1 &1 &0 &0 &1 &0 \1 &0 &0 &0 &0 &1 end{bmatrix}] and determine all code words.
[H= egin{bmatrix} 1&0 &1 &1 &0 &0 \1 &1 &0 &0 &1 &0 \1 &0 &0 &0 &0 &1 end{bmatrix}] and determine all code words.
7 M
8(a)
Let \[E : Z\frac{M}{2} \ \rightarrow Z\frac{n}{2}\] be an encoding function given by a generator matrix G or the ass parity check matrix H then prove that \[C = E :(Z\frac{M}{2})\] is a group code.
6 M
8(b)
i) Define subring and ideal
ii) \[IF \ A - \left \{ \begin{bmatrix} a &o \\ b &c \end{bmatrix}a,b,c,e,z \right \}\] be the subset of the ring R =\[M_{2}(z)\] then prove that A is a subring but not ideal.
ii) \[IF \ A - \left \{ \begin{bmatrix} a &o \\ b &c \end{bmatrix}a,b,c,e,z \right \}\] be the subset of the ring R =\[M_{2}(z)\] then prove that A is a subring but not ideal.
7 M
8(c)
i) Prove that \[z_{n}\] is a field if and only if n is a prime.
ii) Prove that in\[z_{n,}[a]\] is a unit if and only if gcd (a,n) = 1
ii) Prove that in\[z_{n,}[a]\] is a unit if and only if gcd (a,n) = 1
7 M
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