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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.

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