12th Maths Monthly Test - 2 ( Discrete Mathematics )

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 2 ( Discrete Mathematics )-Aug 2020

Time : 2.30 Hours

Part - A

Generic 2 - Answer All Question

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!

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

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

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

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

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

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

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

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

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.

Establish the equivalence property p ➝ q ≡ ㄱp ν q

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

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

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

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

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

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 AΛB

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 ∧ ¬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

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.

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 whether ∗ is a binary operation on the sets given below.
a*b=min (a,b) on A={1,2,3,4,5)
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.’

Part - B

Generic 5 - Answer All Question

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

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

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}

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

Identify the valid statements from the following sentences.

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.

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

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

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12th Maths Monthly Test - 2 ( Discrete Mathematics )

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 2 ( Discrete Mathematics )-Aug 2020

Part - A

Generic 2 - Answer All Question

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!

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

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

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

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

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

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

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

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.

Establish the equivalence property p ➝ q ≡ ㄱp ν q

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

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

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

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

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 ∧ ¬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

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.

Determine whether ∗ is a binary operation on the sets given below. a*b=min (a,b) on A={1,2,3,4,5)

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.’

Part - B

Generic 5 - Answer All Question

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

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

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.

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}

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

Identify the valid statements from the following sentences.

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.

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

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

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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!

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

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

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))

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

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

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

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

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.

Determine whether ∗ is a binary operation on the sets given below.
a*b=min (a,b) on A={1,2,3,4,5)

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.’

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

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

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.

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}

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.

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