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12th Maths Weekly Test - 2 (Discrete Mathematics)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test - 2 (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 5 - Answer All Question

Identify the valid statements from the following sentences.

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

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}

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

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.

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

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