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#### Discrete Mathematics Model Question Paper 1

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. Subtraction is not a binary operation in

(a)

R

(b)

Z

(c)

N

(d)

Q

2. Which one of the following statements has truth value F?

(a)

Chennai is in India or $\sqrt 2$ is an integer

(b)

Chennai is in India or $\sqrt 2$ is an irrational number

(c)

Chennai is in China or $\sqrt 2$ is an integer

(d)

Chennai is in China or $\sqrt 2$ is an irrational number

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. The Identity element of $\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) \right\}$ |x$\in$R, x≠0} under matrix multiplication is

(a)

$\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right)$

(b)

$\left( \begin{matrix} \frac { 1 }{ 4x } & \frac { 1 }{ 4x } \\ \frac { 1 }{ 4x } & \frac { 1 }{ 4x } \end{matrix} \right)$

(c)

$\left( \begin{matrix} \frac { 1 }{ 2 } & \frac { 1 }{ 2 } \\ \frac { 1 }{ 2 } & \frac { 1 }{ 2 } \end{matrix} \right)$

(d)

$\left( \begin{matrix} \frac { 1 }{ 2x } & \frac { 1 }{ 2x } \\ \frac { 1 }{ 2x } & \frac { 1 }{ 2x } \end{matrix} \right)$

5. Which of the following is a contradiction?

(a)

p v q

(b)

p ∧ q

(c)

q v ~ q

(d)

q ∧ ~ q

6. 5 x 2 = 10
7. How many rows are needed for following statement formulae?
(( p ∧ q) ∨ (¬r ∨¬s)) ∧ (¬ t ∧ v))

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

9. Construct the truth table for the following statements.
​​​​​​¬p ∧ ¬q

10. Show that p v (q ∧ r) is a contingency.

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 Z.

14. Define an operation∗ on Q as follows:  a*b=$\left( \frac { a+b }{ 2 } \right)$; a,b ∈Q. Examine the existence of identity and the existence of inverse for the operation * on Q.

15. Show that p ➝ q and q ➝ p are not equivalent

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. 4 x 5 = 20
19. Establish the equivalence property connecting the bi-conditional with conditional: p ↔️ q ≡ (p ➝ q) ∧ (q⟶ p)

20. Using the equivalence property, show that p ↔️ q ≡ ( p ∧ q) v (ㄱp ∧ ㄱq)

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

22. In (N, *) where * is defined by x * y = max (x, y) check the closure axion and identity anion.