1. (a) Define transitive relation. Give an example.

(b) Write the initial functions.

(c) Write distributive lattices.

(d) What is POSET?

(e) If A is finite set with |A|=4,determine how many binary operations can be defined on A?How many of these are commutative?

(f) Define finite set machine.

(g)  What is a partially ordered set?

2. (a) Explain various properties of relations.

(b) When do we say that a function is primitive recursive?

3. (a) If <S1,*> and <S2,*> are monoids having e1 and e2 as the respective identity elements.Prove that the direct product S1*S2 is a monoid with (e1,e2) as the identity elament.

(b) Let G be the set of all non-zero real numbers and let a*b=½ab.Show that <G,*> is an abelian group.

4. (a) Show that subset of linearly ordered poset is sublattice.

(b) Let S={a,b,c}. Draw the diagram of <P(S),>.

5. (a) Expand the function f(w,x,y,z)=w+y`z+x`y into their canonical sum-of-product.

(b)   Simplify the following expression

(a`*b`*c`)+(a*b`*c)+(a*b*c`).

6. (a) Explain homomorphism and isomorphism.

(b) Prove that a code can correct all combinations of k or fewer errors if and only if the minimum distance between the any two code words is at least 2k+1.

7. (a) Describe turing machine with suitable examples.

(b) Write deterministic finite automata for the language contains even number of zeros and even number of ones over the alphabet {0,1}.

8. (a) Write down the Hasse diagram for the positive divisors of 45.

(b) How do you equivalence FSM and sequential circuits? Explain.

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(a) Define transitive relation. Give an example. (b) Write the initial functions. (c) Write distributive lattices. (d) What is POSET? (e) If A is finite set with |A|=4,determine how many binary operations can be defined on A?How many of these are commutative? (f) Define finite set machine. (g)  What is a partially ordered set? 2. (a)...