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12th Maths Monthly Test - 1 ( Discrete Mathematics )

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 1 ( Discrete Mathematics )-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

Which one is the inverse of the statement (PVq)➝(pΛq)?

(p∧q)➝(pVq)
ᄀ(pvq)➝(p∧q)
(ᄀpvᄀq)➝(ᄀp∧ᄀq)
(ᄀp∧ᄀq)➝(ᄀpVᄀq)

Which one of the following statements has truth value F?

Chennai is in India or \(\sqrt{2}\) is an integer
Chennai is in India or \(\sqrt{2}\) is an irrational number
Chennai is in China or \(\sqrt{2}\) is an integer
Chennai is in China or √2 is an irrational number

p

q

(p ∧ q) ⟶ ¬q

T

T

(a)

T

F

(b)

F

T

(c)

F

F

(d)

Which one of the following is correct for the truth value of (p ∧ q)⟶ ¬p?

(a)

(b)

(c)

(d)

T

T

T

T
(a)

(b)

(c)

(d)

F

T

T

T
(a)

(b)

(c)

(d)

F

F

T

T
(a)

(b)

(c)

(d)

T

T

T

F

Which one of the following statements has the truth value T?

sin x is an even function
Every square matrix is non-singular
The product of complex number and its conjugate is purely imaginary
\(\sqrt{5}\) is an irrational number

Determine the truth value of each of the following statements:
(a) 4+2=5 and 6+3=9
(b) 3+2=5 and 6+1=7
(c) 4+5=9 and1+2= 4
(d) 3+2=5 and 4+7=11

(a)

(b)

(c)

(d)

F

T

F

T
(a)

(b)

(c)

(d)

T

F

T

F
(a)

(b)

(c)

(d)

T

T

F

F
(a)

(b)

(c)

(d)

F

F

T

T

Which one is the contrapositive of the statement (pVq)⟶r?

ㄱr➝(ㄱp∧ㄱq)
ㄱr⟶(p∨q)
r⟶(p∧q)
p⟶(q∨r)

Which one of the following is incorrect? For any two propositions p and q, we have

¬ (p∨q) ≡ ¬ p ∧ ¬q
¬ ( p∧ q)≡¬p ∨ ¬q
¬ (p ∨ q)≡¬p∨¬q
¬(¬p)≡ p

A binary operation on a set S is a function from

S ⟶ S
(SxS) ⟶ S
S⟶ (SxS)
(SxS) ⟶ (SxS)

Subtraction is not a binary operation in

R
Z
N
Q

The truth table for (p ∧ q) ∨ ¬q is given below

p

q

(p ∧ q) ∨ (¬q)

T

T

(a)

T

F

(b)

F

T

(c)

F

F

(d)

Which one of the following is true?

(a)

(b)

(c)

(d)

T

T

T

T
(a)

(b)

(c)

(d)

T

F

T

T
(a)

(b)

(c)

(d)

T

T

F

T
(a)

(b)

(c)

(d)

T

F

F

F

If a*b= \(\sqrt{a^2+b^2}\)on the real numbers then * is
commutative but not associative

associative but not commutative

both commutative and associative

neither commutative nor associative

Which one of the following is not true?

Negation of a negation of a statement is the statement itself
If the last column of the truth table contains only T then it is a tautology.
If the last column of its truth table contains only F then it is a contradiction
If p and q are any two statements then p↔ q is a tautology.

In the last column of the truth table for ¬( p ∨ ¬q) the number of final outcomes of the truth value 'F' are

1
2
3
4

The dual of ᄀ(p V q) V [p V (p ∧ ᄀr)] is

ᄀ(p ∧ q) ∧ [p V (p ∧ ᄀr)]
(p ∧ q) ∧ [p ∧ (p V ᄀr)]
ᄀ(p ∧ q) ∧ [p ∧ (p ∧ r)]
ᄀ(p ∧ q) ∧ [p ∧ (pV ᄀr)]

The proposition p ∧ (¬p ∨ q) is

a tautology
a contradiction
logically equivalent to p ∧ q
logically equivalent to p ∨ q

If a compound statement involves 3 simple statements, then the number of rows in the truth table is

9
8
6
3

Part - B

Generic 2 - Answer All Question

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

Construct the truth table for the following statements. (¬p ⟶ r) ∧ ( p ↔ q)
Consider p→q : If today is Monday, then 4 + 4 = 8.

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

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!

How many rows are needed for following statement formulae?
(( p ∧ q) ∨ (¬r ∨¬s)) ∧ (¬ t ∧ v))

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 (A∧B)∨C

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

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

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 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 (A∨B)∧C

Construct the truth table for the following statements.¬(p ∧ ¬q)
Determine whether ∗ is a binary operation on the sets given below.
a*b=min (a,b) on A={1,2,3,4,5)

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^B.

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

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

Part - C

Generic 3 - Answer All Question

Prove that p➝(¬q V r) ≡ ¬pV(¬qVr) using truth table.

Show that p ➝ q and q ➝ p are not equivalent
Verify whether the following compound propositions are tautologies or contradictions or contingency
( p ⟶ q) ↔ (~p ⟶ q)

Show that
¬( p ∧ q) ≡ ¬p V ¬q

Define an operationon Q as follows: ab=\(\left(\frac{a+b}{2} \right)\) ; a,b ∈Q. Examine the closure, commutative, and
associative properties satisfied by*on Q.

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

Write down the
(i) conditional statement
(ii) converse statement
(iii) inverse statement, and
(iv) contrapositive statement for the two statements p and q given below.
p: The number of primes is infinite.
q: Ooty is in Kerala.

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

Define an operation∗ on Q as follows: a*b=\(\left(\frac{a+b}{2} \right)\); a,b ∈Q. Examine the existence of identity and the
existence of inverse for the operation * on Q.
On Z, define ⊗by (m⊗n)=mn+nm: ∀m, n∈Z. Is ⊗binary on Z?

Show that q ➝ p ≡ ¬p ➝ ¬q

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

Part - D

Generic 5 - Answer All Question

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.

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

Using truth table check whether the statements ¬(p V q) V (¬p ∧ q) and ¬p are logically equivalent.
Verify
(i) closure property,
(ii) commutative property,
(iii) associative property,
(iv) existence of identity, and
(v) existence of inverse for following operation on the given set
m*n=m+n-mn; m,n ∈Z

Let M = \(\left\{ \begin{pmatrix} x&x\\x&x \end{pmatrix}: x ∈ R − \{0\} \right\} \)and let ∗ be the matrix multiplication. Determine whetherM is
closed under ∗ . If so, examine the existence of identity, existence of inverse properties for the
operation ∗ on M.

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