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#### Discrete Mathematics Model Question Paper

12th Standard EM

Reg.No. :
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Maths

Time : 00:45:00 Hrs
Total Marks : 40
5 x 1 = 5
1. A binary operation on a set S is a function from

(a)

S ⟶ S

(b)

(SxS) ⟶ S

(c)

S⟶ (SxS)

(d)

(SxS) ⟶ (SxS)

2. In the set Q define a⊙b= a+b+ab. For what value of y, 3⊙(y⊙5)=7?

(a)

y=$\frac{2}{3}$

(b)

y=$\frac{-2}{3}$

(c)

y=$\frac{-3}{2}$

(d)

y=4

3. Which one is the contrapositive of the statement (pVq)⟶r?

(a)

ㄱr➝(ㄱp∧ㄱq)

(b)

ㄱr⟶(p∨q)

(c)

r⟶(p∧q)

(d)

p⟶(q∨r)

4. Which of the following is a tautology?

(a)

p ν q

(b)

p ៱ q

(c)

q v ~ q

(d)

q ៱ ~ q

5. The identity element in the group {R - {1},x} where a * b = a + b - ab is

(a)

0

(b)

1

(c)

$\frac { 1 }{ a-1 }$

(d)

$\frac { a }{ a-1 }$

6. 5 x 2 = 10
7. Let A =$\begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix},B=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$be any two boolean matrices of the same type. Find AvB and A^B.

8. Let p: Jupiter is a planet and q: India is an island be any two simple statements. Give
verbal sentence describing each of the following statements.
(i) ¬p
(ii) p ∧ ¬q
(iii) ¬p ∨ q
(iv) p➝ ¬q
(v) p↔q

9. In the set of integers under the operation * defined by a * b = a + b - 1. Find the identity element.

10. Let S be the set of positive rational numbers and is defined by a * b =$\frac{ab}{2}$. Then find the identity element and the inverse of 2.

11. Let G = {1, w, w2) where w is a complex cube root of unity. Then find the universe of w2. Under usual multiplication.

12. 5 x 3 = 15
13. Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation + on
Ze = the set of all even integers

14. Determine whether ∗ is a binary operation on the sets given below.
(A*v)=a√b is binary on R

15. Let $A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right)$be any three boolean matrices of the same type.
Find (A∨B)∧C

16. In (z, *) where * is defined by a * b = ab, prove that * is not a binary operation on z.

17. Construct the truth table for (-p) v (q ∧ r)

18. 2 x 5 = 10
19. How many rows are needed for following statement formulae?
p ∨ ¬ t ( p ∨ ¬s)

20. How many rows are needed for following statement formulae?
(( p ∧ q) ∨ (¬r ∨¬s)) ∧ (¬ t ∧ v))