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Discrete Mathematics Three Marks Questions

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 45
15 x 3 = 45
1. 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.

2. 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
Ze = the set of all even integers

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
Zo = the set of all even integers

4. Verify
(i) closure property
(ii) commutative property, and
(iii) associative property of the following operation on the given set.
(a*b) = ab;∀a,b∈N (exponentiation property)

5. Determine whether ∗ is a binary operation on the sets given below.
a*b=b=a.|b| on R

6. Determine whether ∗ is a binary operation on the sets given below.
a*b=min (a,b) on A={1,2,3,4,5)

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

8. On Z, define ⊗by (m⊗n)=mn+nm: ∀m, n∈Z. Is ⊗binary on Z?

9. Let *be defined on R by (a*b)=a+b+ab-7. is*binary on R? If so, find 3$\left( \frac { -7 }{ 15 } \right)$.

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

11. Consider the binary operation ∗ defined on the set A = {a,b,c, d} by the following table:

 * a b c d a a c b d b c d a a d d b a c

Is it commutative and associative?

12. 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 AVB

13. 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 AΛB

14. 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

15. 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