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PART D
Answer any two full questions, each carries9 marks.
State and prove Lagrange’s Theorem

For any Boolean algebra 8, prove that a+b = 840 and ab = ac => b = for alla, b,
c €B

Check whether the following is distributive lattice or not

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If (a+b)?=a?+2ab-+b’V a,b,c €R, Prove that R is a commutative Ring and conversely
Prove that the set G={0,1,2,3,4,5} is a abelian group under addition modulo 6.
Define lattice homomorphism.

PART E

Answer any four full questions, each carries10 marks.
Show that n*+2n is divisible by 3 by mathematical induction

Show that the following premises are inconsistent

If Jack misses many classes through illness, then he fails high school
If Jack fails high school then he is uneducated
If Jack reads a lot of books, then he is not educated

4. Jack misses many class through illness and reads a lot of books
Construct truth table for the following formula (P VQ) v(~P VQ) v(P A~Q)

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Show that R->S can be derived from the premises P->(Q->S), ~R VP and Q
Show that (x)(P(x) VQ(x))=>(x)P(x) V(Ax)Q(x)
Explain proof by contradiction with an example

Without using truth table prove that

(P VQ) A~(~P. A(~Q V~R))) V(~P A~Q) V(~P A ~R) is a tautology
Show that the conclusion C follows from the premises H1,H2
H1:~Q H2:P->Q C:~P

Symbolize the expression “All the world loves a lover”

Prove that 5+10+15.....4+5n=5n(n+1)/2using mathematical induction

Determine the validity of the following arguments

Every living thing is a plant or animal

John’s gold fish is alive and it is not a plant

All animals have hearts

Therefore, John’s gold fish has heart.

Show that RA(P VQ)is a valid conclusion from the premises P VQ,Q->R,P->M, and
~M

KKK

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