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Discrete Mathematics Two Marks Questions

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 45
    20 x 2 = 40
  1. Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
    a*b = a + 3ab − 5b2; ∀a,b∈Z

  2. Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
    \(a*b=\left( \frac { a-1 }{ b-1 } \right) ,\forall a,b\in Q\)

  3. 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.

  4. 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\(\wedge\)B.

  5. 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

  6. Determine the truth value of each of the following statements
    (i) If 6 + 2 = 5 , then the milk is white.
    (ii) China is in Europe or \(\sqrt3\) is an integer
    (iii) It is not true that 5 + 5 = 9 or Earth is a planet
    (iv) 11 is a prime number and all the sides of a rectangle are equal

  7. Fill in the following table so that the binary operation ∗ on A = {a, b, c} is commutative.

    * a b c
    a b    
    b c b a
    c a   c
  8. Write the converse, inverse, and contrapositive of each of the following implication.
    If x and y are numbers such that x = y, then x2 = y2

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

  10. Construct the truth table for the following statements.
    ( p V q) V ¬q

  11. Construct the truth table for the following statements.
    (¬p ⟶ r) ∧ ( p ↔️ q)

  12. Verify whether the following compound propositions are tautologies or contradictions or contingency
    (p ∧ q) ∧ ¬ (p ∨ q)

  13. Verify whether the following compound propositions are tautologies or contradictions or contingency
    (( p V q)∧ ¬ p) ➝ q

  14. Verify whether the following compound propositions are tautologies or contradictions or contingency
    ( p ⟶ q) ↔️ (~ p ⟶ q)

  15. Verify whether the following compound propositions are tautologies or contradictions or contingency
    ((p⟶ q) ∧ (q ⟶ r)) ⟶ (p ⟶ r)

  16. Show that ¬( p ∧ q) ≡ ¬p V ¬q

  17. Prove that q ➝ p ≡ ¬p ➝ ¬q

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

  19. Using truth table check whether the statements ¬(p V q) V (¬p ∧ q) and ¬p are logically equivalent.

  20. Prove p⟶(q⟶r) ☰ (p ∧ q)⟶r without using truth table.

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