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12th Maths Weekly Test - 3 (Discrete Mathematics)

St. Britto Hr. Sec. School - Madurai

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

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

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

Which one of the following is a binary operation on N?

Subtraction
Multiplication
Division
All the above

In the set R of real numbers ‘*’ is defined as follows. Which one of the following is not a binary operation on R?

a*b=min (a.b)
a*b= max (a,b)
a*b= a
a*b=a^b

The operation * defined by a*b = \(\frac{ab}{7}\) is not a binary operation on

Q⁺
Z
R
C

In the set Q define a⊙b= a+b+ab. For what value of y, 3⊙(y⊙5)=7?

y = \(\frac{2}{ 3}\)
y = \(\frac{-2}{ 3}\)
y =\(\frac{-3}{ 2}\)
y = 4

If ab= \(\sqrt{a^2+b^2}\)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 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

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

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

9
8
6
3

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

ㄱr➝(ㄱp∧ㄱq)
ㄱr⟶(p∨q)
r⟶(p∧q)
p⟶(q∨r)

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

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

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

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

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

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 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.

Part - B

Generic 2 - Answer All Question

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 − 5b ;∀a,b∈Z

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.’

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

Consider p→q : If today is Monday, then 4 + 4 = 8.

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

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

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

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

Part - C

Generic 3 - Answer All Question

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.

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

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)

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

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

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.

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

Verify whether the following compound propositions are tautologies or contradictions or contingency
( p ⟶ q) ↔ (~p ⟶ q)

Show that q ➝ p ≡ ¬p ➝ ¬q

Show that ¬(p↔ q) ≡ p↔ ¬q

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