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12th Maths Weekly Test -1 (Discrete Mathematics)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -1 (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)

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

Q⁺
Z
R
C

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 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

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 is a binary operation on N?
Subtraction

Multiplication

Division

All the above

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

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

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

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.’

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

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

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)

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

Part - D

Generic 5 - Answer All Question

Identify the valid statements from the following sentences.

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

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