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#### Discrete Mathematics Five Marks Questions

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
10 x 5 = 50
1. Write the statements in words corresponding to ¬p, p ∧ q , p ∨ q and q ∨ ¬p, where p is ‘It is cold’ and q is ‘It is raining.’

2. How many rows are needed for following statement formulae?
p ∨ ¬ t ( p ∨ ¬s)

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

4. Construct the truth table for $(p\overset { \_ \_ }{ \vee } q)\wedge (p\overset { \_ \_ }{ \vee } \neg q)$

5. Verify
(i) closure property,
(ii) commutative property,
(iii) associative property,
(iv) existence of identity, and
(v) existence of inverse for the operation +5 on Z5 using table corresponding to addition modulo 5.

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

7. Define an operation*on Q as follows: a*b=$\left( \frac { a+b }{ 2 } \right)$; a,b ∈Q. Examine the closure, commutative, and associative properties satisfied by*on Q.

8. Let M=$\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\}$ and let * be the matrix multiplication. Determine whether M is closed under ∗. If so, examine the commutative and associative properties satisfied by ∗ on M.

9. Let M=$\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\}$ and let ∗ be the matrix multiplication. Determine whetherM is closed under ∗ . If so, examine the existence of identity, existence of inverse properties for the operation ∗ on M.

10. Let A be Q\{1}. Define ∗ on A by x*y = x + y − xy . Is ∗ binary on A? If so, examine the existence of identity, existence of inverse properties for the operation ∗ on A.