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Discrete Mathematics - Two Marks Study Materials

12th Standard EM

    Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 30
    15 x 2 = 30
  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. How many rows are needed for following statement formulae?
    p ∨ ¬ t ( p ∨ ¬s)

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

  5. Let A={a+\(\sqrt5\)b:a,b∈Z}. Check whether the usual multiplication is a binary operation on A.

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

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

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

  10. 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 

  11. Show that p v (~p) is a tautology.

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

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

  14. 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.

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

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