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12th Maths Monthly Test - 3 ( Discrete Mathematics )

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 3 ( Discrete Mathematics )-Aug 2020

Time : 2.35 Hours

Part - A

Generic 2 - Answer All Question

Let A = \(\begin{pmatrix} 1\quad0 &1\quad0\\0\quad1&0\quad1\\1\quad0&0\quad1 \end{pmatrix}\),B = \(\begin{pmatrix}0\quad1&0\quad1\\ 1\quad0 &1\quad0\\1\quad0&0\quad1 \end{pmatrix}\),C = \(\begin{pmatrix}1\quad1&0\quad1\\ 0\quad1 &1\quad0\\1\quad1&1\quad1 \end{pmatrix}\)be any three boolean matrices of the same type. Find AVB

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

Write each of the following sentences in symbolic form using statement variables p and q.
(i) 19 is not a prime number and all the angles of a triangle are equal.
(ii) 19 is a prime number or all the angles of a triangle are not equal
(iii) 19 is a prime number and all the angles of a triangle are equal
(iv) 19 is not a prime number

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 \(\sqrt{3}\) 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

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

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

Write the converse, inverse, and contrapositive of each of the following implication.
(i) If x and y are numbers such that x = y, then x²= y²
(ii) If a quadrilateral is a square then it is a rectangle.

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

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

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

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.’

Which one of the following sentences is a proposition?
(i) 4 + 7 =12
(ii) What are you doing?
(iii) 3ⁿ ≤ 8,1 n ∈ N
(iv) Peacock is our national bird
(v) How tall this mountain is!

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

Determine whether ∗ is a binary operation on the sets given below.
(A*v)=a√b is binary on R
How many rows are needed for following statement formulae?
(( p ∧ q) ∨ (¬r ∨¬s)) ∧ (¬ t ∧ v))

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

Establish the equivalence property p ➝ q ≡ ㄱp ν q

Part - B

Generic 3 - Answer All Question

Show that q ➝ p ≡ ¬p ➝ ¬q

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

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

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

Check whether the statement p➝(q➝p) is a tautology or a contradiction without using the truth
table.

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 d a b c

c

c

d

a

a

d

d

b

a

c

Is it commutative and associative?

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.

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

Define an operation∗ on Q as follows: ab=\(\left(\frac{a+b}{2} \right)\); a,b ∈Q. Examine the existence of identity and the
existence of inverse for the operation on Q.

Construct the truth table for (p \(\bar { ∨ } \) q) ∧ (p\(\bar { ∨ } \) ¬q)

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_e = the set of all even integers

Part - C

Generic 5 - Answer All Question

Using the equivalence property, show that p ↔ q ≡ ( p ∧ q) v (ㄱp ∧ ㄱq)

Establish the equivalence property connecting the bi-conditional with conditional: p ↔ q ≡ (p ➝ q)
∧ (q⟶ p)

Let A be Q\{1}. Define ∗ on A by x*y = x + y − xy . Is ∗ binary on A? If so, examine the commutative
and associative properties satisfied by ∗ on A.

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

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.
Let M = \(\left\{ \begin{pmatrix} x&x\\x&x \end{pmatrix}: x ∈ R − \{0\} \right\} \)and let * be the matrix multiplication. Determine whether M is
closed under ∗. If so, examine the commutative and associative properties satisfied by ∗ on M

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.

Verify
(i) closure property,
(ii) commutative property,
(iii) associative property,
(iv) existence of identity, and
(v) existence of inverse for the operation ×11 on a subset A = {1,3,4,5,9}
of the set of remainders {0,1,2,3,4,5,6,7,8,9,10}

Identify the valid statements from the following sentences.

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