Samacheer Kalvi solutions for Mathematics - Volume 1 and 2 [English] Class 12 TN Board chapter 12 - Discrete Mathematics [Latest edition]

Determine whether * is a binary operation on the sets-given below.

a * b – a.|b| on R

Determine whether * is a binary operation on the sets-given below.

a * b = min (a, b) on A = {1, 2, 3, 4, 5}

Determine whether * is a binary operation on the sets-given below.

(a * b) = `"a"sqrt("b")` is binary on R

On Z, define * by (m * n) = mⁿ + n^m : ∀m, n ∈ Z Is * binary on Z?

Let * be defined on R by (a * b) = a + b + ab – 7. Is * binary on R? If so, find 3 * `((-7)/15)`

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

Define an operation * on Q as follows: a * b = `(("a" + "b")/2)`; a, b ∈ Q. Examine the closure, commutative and associate properties satisfied by * on Q.

Define an operation * on Q as follows: a * b = `(("a" + "b")/2)`; a, b ∈ Q. Examine the existence of identity and the existence of inverse for the operation * on Q.

Fill in the following table so that the binary operation * on A = {a, b, c} is commutative.

Consider the binary operation * defined on the set A = {a, b, c, d} by the following table:

Is it commutative and associative?

Let A = `((1, 0, 1, 0),(0, 1, 0, 1),(1, 0, 0, 1))`, B = `((0, 1, 0, 1),(1, 0, 1, 0),(1, 0, 0, 1))`, C = `((1, 1, 0, 1),(0, 1, 1, 0),(1, 1, 1, 1))` be any three boolean matrices of the same type. Find A v B

Let A = `((1, 0, 1, 0),(0, 1, 0, 1),(1, 0, 0, 1))`, B = `((0, 1, 0, 1),(1, 0, 1, 0),(1, 0, 0, 1))`, C = `((1, 1, 0, 1),(0, 1, 1, 0),(1, 1, 1, 1))` be any three boolean matrices of the same type. Find A ∧ B

Let A = `((1, 0, 1, 0),(0, 1, 0, 1),(1, 0, 0, 1))`, B = `((0, 1, 0, 1),(1, 0, 1, 0),(1, 0, 0, 1))`, C = `((1, 1, 0, 1),(0, 1, 1, 0),(1, 1, 1, 1))` be any three boolean matrices of the same type. Find (A v B) ∧ C

Let A = `((1, 0, 1, 0),(0, 1, 0, 1),(1, 0, 0, 1))`, B = `((0, 1, 0, 1),(1, 0, 1, 0),(1, 0, 0, 1))`, C = `((1, 1, 0, 1),(0, 1, 1, 0),(1, 1, 1, 1))` be any three boolean matrices of the same type. Find (A ∧ B) v C

Let M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}` and let * be the matrix multiplication. Determine whether M is closed under *. If so, examine the commutative and associative properties satisfied by * on M

Let M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}` and let * be the matrix multiplication. Determine whether M is closed under * . If so, examine the existence of identity, existence of inverse properties for the operation * on M

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

Let A be Q\{1}. Define * on A by x * y = x + y – xy. Is * binary on A? If so, examine the existence of an identity, the existence of inverse properties for the operation * on A

Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing the following statement.

¬ P

Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing the following statement.

P ∧ ¬q

Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing the following statement.

¬p v q

Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing the following statement.

p → ¬q

Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing the following statement.

p ↔ q

Write the following sentences in symbolic form using statement variables p and q.

19 is not a prime number and all the angles of a triangle are equal

Write the following sentences in symbolic form using statement variables p and q.

19 is a prime number or all the angles of a triangle are not equal

Write the following sentences in symbolic form using statement variables p and q.

19 is a prime number and all the angles of a triangle are equal

Write the following sentences in symbolic form using statement variables p and q.

19 is not a prime number

Determine the truth value of the following statement.

If 6 + 2 = 5, then the milk is white.

Determine the truth value of the following statement.

China is in Europe dr `sqrt(3)` is art integer

Determine the truth value of the following statement.

It is not true that 5 + 5 = 9 or Earth is a planet

Determine the truth value of the following statement.

11 is a prime number and all the sides of a rectangle are equal

Which one of the following sentences is a proposition?

4 + 7 = 12

Which one of the following sentences is a proposition?

What are you doing?

Which one of the following sentences is a proposition?

3ⁿ ≤ 81, n ∈ N

Which one of the following sentences is a proposition?

Peacock is our national bird

Which one of the following sentences is a proposition?

How tall this mountain is!

Write the converse, inverse, and contrapositive of the following implication.

If x and y are numbers such that x = y, then x² = y²

Write the converse, inverse, and contrapositive of the following implication.

If a quadrilateral is a square then it is a rectangle

Construct the truth table for the following statement.

¬P ∧ ¬q

Construct the truth table for the following statement.

¬(P ∧ ¬q)

Construct the truth table for the following statement.

(p v q) v ¬q

Construct the truth table for the following statement

(¬p → r) ∧ (p ↔ q)

Verify whether the following compound propositions are tautologies or contradictions or contingency.

(p ∧ q) ∧¬ (p v q)

Verify whether the following compound propositions are tautologies or contradictions or contingency.

((p v q) ∧¬p) → 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) ∧ (q → r)) → (p → r)

Show that (p ∧ q) ≡ ¬p v ¬q

Show that ¬(p → q) ≡ p ∧¬q

Prove that q → p ≡ ¬p → ¬q

Show that p → q and q → p are not equivalent

Show that ¬(p ↔ q) ≡ p ↔ ¬q

Check whether the statement p → (q → p) is a tautology or a contradiction without using the truth table

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

Prove p → (q → r) ≡ (p ∧ q) → r without using the truth table

Prove that p → (¬q v r) ≡ ¬p v (¬q v r) using truth table

Choose the correct alternative:

A binary operation on a set S is a function from

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Subtraction is not a binary operation in

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

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

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The operation * defined by a * b = `"ab"/7` is not a binary operation on

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

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If a * b = `sqrt("a"^2 + "b"^2)` on the real numbers then * is

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

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Which one of the following statements has truth value F?

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

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Which one is the inverse of the statement (p v q) → (p ∧ q)?

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Which one is the contrapositive of the statement (p v q) → r?

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

Which one of the following is true?

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In the last column of the truth table for ¬(p v ¬q) the number of final outcomes of the truth value ‘F’ is

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

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

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The dual of ¬(p v q) v [p v(p ∧ ¬r)] is

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The proposition p∧(¬p∨q)] is

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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 and 1 + 2 = 4
(d) 3 + 2 = 5 and 4 + 7 = 11

Choose the correct alternative:

Which one of the following is not true?