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#### Discrete Mathematics - Two Marks Study Materials

12th Standard EM

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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.