Answer the following questions in short.

Question 11 Mark

Write the truth values of the following statements :

Ɐ n ∈ N, n + 6 > 8.

Answer

Ɐ n ∈ N, 11 + 6 > 8 is a false statement, hence its truth value is F.
{n = 1 ∈ N, n = 2 ∈ N do not satisfy n + 6 > 8).

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Question 21 Mark

Write the truth values of the following statements :

In ∆ ABC if all sides are equal then its all angles are equal.

Answer

Let p : ABC is a triangle and all its sides are equal.
q : Its all angles are equal.
Then the symbolic form of the given statement is p → q
If the truth value of p is T, then the truth value of q is T.
∴ the truth value of p → q is T. … [T → T ≡ T].

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Question 31 Mark

Write the truth values of the following statements :

The square of any even number is odd or the cube of any odd number is odd.

Answer

Let p : The square of any even number is odd.
q : The cube of any odd number is odd.
Then the symbolic form of the given statement is p ∨ q.
The truth values of p and q are F and T respectively.
∴ the truth value of p ∨ q is T. … [F ∨ T ≡ T].

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Question 41 Mark

Write the truth values of the following statements :

Ǝ n ∈ N such that n + 5 > 10.

Answer

Ǝ n ∈ N, such that n + 5 > 10 is a true statement, hence its truth value is T.
(All n ≥ 6, where n ∈ N, satisfy n + 5 > 10).

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Question 51 Mark

Write the truth values of the following statements :
$\forall n \in N, n^2+n$ is even number while $n ^2- n$ is an odd number.

Answer

Let $p: \forall n \in N, n^2+n$ is an even number.
$q : \forall n \in N , n ^2- n$ is an odd number.
Then the symbolic form of the given statement is $p \wedge q$.
The truth values of $p$ and $q$ are $T$ and $F$ respectively.
$\therefore$ the truth value of $p \wedge q$ is $F \ldots[T \wedge F \equiv F]$.

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Question 61 Mark

Write the truth values of the following statements :$\sqrt{5}$ is an irrational but $3 \sqrt{5}$ is a complex number.

Answer

Let $p: \sqrt{5}$ is an irrational.

$q: 3 \sqrt{5}$ is a complex number.

Then the symbolic form of the given statement is $p \wedge q$.

The truth values of $p$ and $q$ are $T$ and $F$ respectively.

$\therefore$ the truth value of $p \wedge q$ is $F . \ldots[T \wedge F \equiv F]$

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Question 71 Mark

Write the negations of the following : Some triangles are equilateral triangle.

Answer

All triangles are not equilateral triangles.

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Question 81 Mark

Write the negations of the following : Ǝ x ∈ A, such that x + 9 ≤ 15.

Answer

Ɐ x ∈ A, x + 9 > 15.

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Question 91 Mark

Write the negations of the following : Ɐ n ∈ A, n + 7 > 6.

Answer

The negation of the given statements are :
Ǝ n ∈ A, such that n + 7 ≤ 6.
OR Ǝ n ∈ A, such that n + 7 ≯ 6.

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Question 101 Mark

Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements :If x is a whole number the x + 6 = 0.

Answer

It is a statement which is false, hence its truth value is ‘F’.

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Question 111 Mark

Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements :
$cos^2\theta – \sin^2\theta = cos2\theta $ for all $θ \in R.$

Answer

It is a statement which is true, hence its truth value is $‘T’.$

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Question 121 Mark

Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements :Please get me a glass of water.

Answer

It is an imperative sentence, hence it is not a statement.

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Question 131 Mark

Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements :India is a country and Himalayas is a river.

Answer

It is a statement which is false, hence its truth value is ‘F’. ….[T ∧ F ≡ F]

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Question 141 Mark

Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements :π is an irrational number.

Answer

It is a statement which is true, hence its truth value is ‘T’.

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Question 151 Mark

Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements :4! = 24.

Answer

It is a statement which is true, hence its truth value is ‘T’.

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Question 161 Mark

Determine the truth values ofp and q in the following cases : (p ∧ q) is F and (p ∧ q) → q is T

Answer

Since the truth values of (p ∧ q) is F and (p ∧ q) → q is T, from the table, the truth values of p and q are either T and F respectively or F and T respectively or both F.

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Question 171 Mark

Determine the truth values ofp and q in the following cases : (p ∨ q) is T and (p ∨ q) → q is F

Answer

Since the truth values of (p ∨ q) is T and (p ∨ q) → q is F, from the table, the truth values of p and q are T and F respectively.

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Question 181 Mark

Determine the truth values ofp and q in the following cases : (p ∨ q) is T and (p ∧ q) is T

Answer

Since p ∨ q and p ∧ q both are T, from the table the truth values of both p and q are T.

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Question 191 Mark

In the following switching circuit,: Write symbolic form

Answer

The symbolic form of given circuit is (p ∧ q) ∨ (~ p ∧ q) ∨ (r ∧ ~ q).

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Question 201 Mark

Without using truth table prove that : $\sim p \wedge q=(p \vee q) \wedge \sim p$

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Question 211 Mark

Without using truth table prove that :$\sim(p \vee q) \vee(\sim p \wedge q)=\sim p$

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Question 221 Mark

Without using truth table prove that : i) $\quad p \leftrightarrow q \equiv \sim(p \wedge \sim q) \wedge \sim(q \wedge \sim p)$

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Question 231 Mark

Rewrite the following statements without using if ...... then : If it is cold, then we wear woolen clothes

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Question 241 Mark

Rewrite the following statements without using if ...... then : If prices increase then the wages rise

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Question 251 Mark

Write the negations of the following stating the rules used : $(p \wedge q) \rightarrow(\sim p \vee r)$

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Question 261 Mark

Write the negations of the following stating the rules used : $(\sim p \wedge q) \vee(p \wedge \sim q)$

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Question 271 Mark

Write the negations of the following stating the rules used : $p \wedge(q \vee r)$

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Question 281 Mark

Write the negations of the following stating the rules used : $(p \rightarrow q) \vee r$

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Question 291 Mark

Write the negations of the following stating the rules used : $(p \vee q) \wedge(q \vee \sim r)$

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Question 301 Mark

Using rules of negation write the negations of the following with justification. (~ p ∨ ~q) ∨ (p ∧ ~q)

Answer

The negation of (~ p ∨ ~q) ∨ (p ∧ ~ q) is
~ [(~p ∨ ~q) ∨ (p ∧ ~ q)]
≡ ~(~p ∨ ~q) ∧ ~(p ∧ ~q) … (Negation of disjunction)
≡ [~(~p) ∧ ~(~q)] ∧ [~p ∨ ~(~q)] … (Negation of disjunction and conjunction)
≡ (p ∧ q) ∧ (~p ∨ q) … (Negation of negation)

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Question 311 Mark

Using rules of negation write the negations of the following with justification. (p ∨ ~q) → (p ∧ ~q)

Answer

The negation of (p ∨ ~q) → (p ∧ ~q) is
~[(p ∨ ~q) → (p ∧ ~q)]
≡ (p ∨ ~q) ∧ ~(p ∧ ~q) … (Negation of implication)
≡ (p ∨ ~q) ∧ [ ~p ∨ ~(~q)] … (Negation of conjunction)
≡ (p ∨ ~q) ∧ (~p ∨ q) … (Negation of negation)

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Question 321 Mark

Using rules of negation write the negations of the following with justification. ~(p ∧ q) ∨ (p ∨ ~q)

Answer

The negation of ~(p ∧ q) ∨ (p ∨ ~q) is
~[~(p ∧ q) ∨ (p ∨ ~q)] ≡ ~[~(p ∧ q)] ∧ ~(p ∨ ~q) … (Negation of disjunction)
≡ ~[~(p ∧ q)] ∧ [ p ∧ ~(~q)] … (Negation of disjunction)
≡ (p ∧ q) ∧ (~ p ∧ q) … (Negation of negation)

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Question 331 Mark

Using rules of negation write the negations of the following with justification. p → (p ∨ ~q)

Answer

Solution:
The negation of p → (p ∨ ~q) is
~ [p → (p ∨ ~q)] ≡ p ∧ ~ (p ∧ ~p) … (Negation of implication)
≡ p ∧ [ ~ p ∧ ~ (~(q)] … (Negation of disjunction)
≡ p ∧ (~ p ∧ q) (Negation of negation)

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Question 341 Mark

Using rules of negation write the negations of the following with justification. (p ∨ ~q) ∧ r

Answer

The negation of (p ∨ ~ q) ∧ r is
~[(p ∨ ~q) ∧ r] ≡ ~(p ∨ ~q) ∨ ~r … (Negation of conjunction)
≡ [ ~p ∧ ~(~q)] ∨ ~ r… (Negation of disjunction)
≡ (~ p ∧ q) ∧ ~ r … (Negation of negation)

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Question 351 Mark

Using rules of negation write the negations of the following with justification. p ∨ ~q

Answer

The negation of p ∨ ~ p is
~ (p ∨ ~(q) ≡ ~p ∧ ~(~(q) … (Negation of disjunction)
≡ ~ p ∧ q … (Negation of negation)

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Question 361 Mark

Using rules of negation write the negations of the following with justification. p ∧ ~q

Answer

The negation of p ∧ ~q is
~(p ∧ ~q) ≡ ~p ∨ ~(~q) … (Negation of conjunction)
≡ ~ p ∨ q … (Negation of negation)

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Question 371 Mark

Using rules of negation write the negations of the following with justification. ~q → p

Answer

The negation of ~q → p is
~(~q → p) ≡ ~ q ∧ ~p…. (Negation of implication)

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Question 381 Mark

Rewrite the following statements without using if .. then.It f(2) = 0 then f(x) is divisible by (x – 2).

Answer

f(2) ≠ 0 or f(x) is divisible by (x – 2).

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Question 391 Mark

Rewrite the following statements without using if .. then.It 2 is a rational number then $\sqrt{2}$ is irrational number.

Answer

It 2 is not a rational number then $\sqrt{2}$ is irrational number.

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Question 401 Mark

Rewrite the following statements without using if .. then.If a man is a judge then he is honest.

Answer

Since p → ≡ ~p ∨ q, the given statements can be written as :
A man is not a judge or he is honest.

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Question 411 Mark

Write the negations of the following.Ǝ x ∈ N such that x – 17 < 20

Answer

The negation of the given statement is :
‘Ɐ x ∈ N, x – 17 ≯ 20.’

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Question 421 Mark

Write the negations of the following.$Ɐ n \in N, n^2 + n + 2$ is divisible by $4$.

Answer

The negation of the given statement is :
‘Ǝ n \in N, such that $n^2 + n + 2$ is not divisible by $4.’$

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Question 431 Mark

Write the negations of the following.All natural numbers are whole numers.

Answer

The negation of the given statement is :
‘Some natural numbers are not whole numbers.’

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Question 441 Mark

Write the negations of the following.$\sqrt{2}$ is a rational number.

Answer

Let $p: \sqrt{2}$ is a rational number.
The negation of the given statement is
' $\sim p: \sqrt{2}$ is not a rational number.'

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Question 451 Mark

Write the negations of the following.If it is raining then we will go and play football.

Answer

Let p : It is raining.
q : We will go.
r : We play football.
Then the symbolic form of the given statement is p → (q ∧ r).
Since ~[p → (q ∧ r)] ≡ p ∧ ~(q ∧ r) ≡ p ∧ (q ∨ ~r), the negation of the given statement is :
‘It is raining and we will not go or not play football.’

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Question 461 Mark

Write the negations of the following.It is cold and raining.

Answer

Let p : It is cold.
q : It is raining.
Then the symbolic form of the given statement is p ∧ q.
Since ~(p ∧ q) ≡ ~p ∨ ~q, the negation of the given statement is :
‘It is not cold or not raining.’

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Question 471 Mark

Write the negations of the following.Qudrilateral is a square if and only if it is a rhombus.

Answer

Let p : Quadrilateral is a square.
q : It is a rhombus.
Then the symbolic form of the given statement is p ↔ q.
Since ~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p), the negation of given statement is :
‘ Quadrilateral is a square but it is not a rhombus or quadrilateral is a rhombus but it is not a square.’

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Question 481 Mark

Write the negations of the following.11 < 15 and 25 > 20

Answer

Let p: 11 < 15, q : 25 > 20.
Then the symbolic form of the given statement is p ∧ q.
Since ~(p ∧ q) ≡ ~p ∨ ~q, the negation of given statement is :
’11 ≮ 15 or 25 > 20.’ OR
’11 ≯ 15 or 25 ≮ 20.’

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Question 491 Mark

Write the negations of the following. x + 8 > 11 or y – 3 = 6

Answer

Let p : x + 8 > 11, q : y — 3 = 6.
Then the symbolic form of the given statement is p ∨ q.
Since ~(p ∨ q) ≡ ~p ∧ ~q, the negation of given statement is :
‘x + 8 > 11 and y – 3 ≠ 6’ OR
‘x + 8 ≮ 11 and y – 3 ≠ 6’

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Question 501 Mark

Write the duals of each of the following. (p ∨ q) ∨ c

Answer

(p ∧ q) ∧ t

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Question 511 Mark

Write the duals of each of the following. ~p ∨ (q ∧ r) ∧ t

Answer

~p ∧ (q ∨ r) ∨ c

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Question 521 Mark

Write the duals of each of the following. c ∨ {p ∧ (q ∨ r)}

Answer

t ∧ {p ∧ (q Ar)}

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Question 531 Mark

Write the duals of each of the following. [~(p ∨ q)] ∧ [p ∨ ~(q ∧ ~s)]

Answer

[ ~(p ∧ q)] ∨ [p ∧ ~(q ∨ ~s)]

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Question 541 Mark

Write the duals of each of the following. ~p ∧ (~q ∧ (p ∨ q) ∧ ~r)

Answer

~p ∨ (~q ∨ (p ∧ q) ∨ ~r)

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Question 551 Mark

Write the duals of each of the following. (~p ∨ q) ∧ (~r ∧ s)

Answer

(~p ∧ q) ∨ (~r ∨ s)

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Question 561 Mark

Write the duals of each of the following. p ∧ ~q

Answer

p ∨ ~q

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Question 571 Mark

Write the duals of each of the following. (p ∨ q) ∧ (r ∨ s)

Answer

(p ∧ q) ∨ (r ∧ s)

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Question 581 Mark

Write the duals of each of the following. p ∧ (q ∧ r)

Answer

p ∨ (q ∨ r)

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Question 591 Mark

Write the duals of each of the following. p ∨ (q ∧ r)

Answer

The duals of the given statement patterns are :
p ∧ (q ∨ r)

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Question 601 Mark

If A = {3, 5, 7, 9, 11, 12}, determine the truth value of each of the following. Ɐ x ∈ A, 2x + 9 > 14

Answer

For each x ∈ A, 2x + 9 > 14. So the given statement is true, hence its truth value is T.

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Question 611 Mark

If A = {3, 5, 7, 9, 11, 12}, determine the truth value of each of the following. Ǝ x ∈ A such that 3x + 8 > 40

Answer

Clearly x = 11 ∈ A and x = 12 ∈ A satisfies 3x + 8 > 40. So the given statement is true, hence its truth value is T.

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Question 621 Mark

If A = {3, 5, 7, 9, 11, 12}, determine the truth value of each of the following. Ɐ x ∈ A, x is an even number

Answer

x = 3 ∈ A, x = 5 ∈ A, x = 7 ∈ A, x = 9 ∈ A, x = 11 ∈ A do not satisfy x is an even number. So the given statement is false, hence its truth value is F.

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Question 631 Mark

If $A = {3, 5, 7, 9, 11, 12}$, determine the truth value of each of the following. $Ǝ x ∈ A$ such that $x^2 < 0$

Answer

There is no $x ∈ A$ which satisfies $x^2 < 0$. So the given statement is false, hence its truth value is $F$.

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Question 641 Mark

If $A=\{3,5,7,9,11,12\}$, determine the truth value of each of the following. $\forall x \in A, x^2+x$ is an even number

Answer

For each $x \in A, x^2+x$ is an even number. So the given statement is true, hence its truth value is $T$.

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Question 651 Mark

If A = {3, 5, 7, 9, 11, 12}, determine the truth value of each of the following. Ǝ x ∈ A such that x – 8 = 1

Answer

Clearly x = 9 ∈ A satisfies x – 8 = 1. So the given statement is true, hence its truth value is T.

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Question 661 Mark

Write the truth values of the following :24 is a composite number or 17 is a prime number.

Answer

Let p : 24 is a composite number.
q : 17 is a prime number.
Then the symbolic form of the given statement is p ∨ q.
The truth values of both p and q are T.
∴ the truth value of p ∨ q is T. … [T ∨ T ≡ T]

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Question 671 Mark

Write the truth values of the following :Milk is white if and only if sky is blue.

Answer

Let p : Milk is white.
q : Sky is blue
Then the symbolic form of the given statement is p ↔ q.
The truth values of both p and q are T.
∴ the truth value of p ↔ q is T. … [T ↔ T ≡ T]

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Question 681 Mark

Write the truth values of the following :If 3 × 5 = 8, then 3 + 5 = 15.

Answer

Let p : 3 × 5 = 8.
q : 3 + 5 = 15.
Then the symbolic form of the given statement is p → q.
The truth values of both p and q are F.
∴ the truth value of p → q is T. … [F → F ≡ T]

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Question 691 Mark

Write the truth values of the following :It is not true that 5 – 3i is a real number.

Answer

Let p : 5 – 3i is a real number.
Then the symbolic form of the given statement is ~ p.
The truth values of p is F.
∴ the truth values of ~ p is T. … [~ F ≡ T]

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Question 701 Mark

Write the truth values of the following :5 is a prime number and 7 divides 94.

Answer

Let p : 5 is a prime number.
q : 7 divides 94.
Then the symbolic form of the given statement is p∧q.
The truth values of p and q are T and F respectively.
∴ the truth value of p ∧ q is F. … [T ∧ F ≡ F]

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Question 711 Mark

Write the truth values of the following :64 is a perfect square and 46 is a prime number.

Answer

Let p : 64 is a perfect square.
q : 46 is a prime number.
Then the symbolic form of the given statement is p∧q.
The truth values of p and q are T and F respectively.
∴ the truth value of p ∧ q is F. … [T ∧ F ≡ F]

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Question 721 Mark

Write the truth values of the following :4 is odd or 1 is prime.

Answer

Let p : 4 is odd.
q : 1 is prime.
Then the symbolic form of the given statement is p∨q.
The truth values of both p and q are F.
∴ the truth value of p v q is F. … [F ∨ F = F]

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Question 731 Mark

Write the negations of the following :7 + 3 > 5.

Answer

7 + 3 > 5.

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Question 741 Mark

Write the negations of the following :Polygon ABCDE is a pentagon.

Answer

Polygon ABCDE is not a pentagon.

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Question 751 Mark

Write the negations of the following :$\sqrt{2}$ is a rational number.

Answer

$\sqrt{2}$ is not a rational number.

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Question 761 Mark

Write the negations of the following :
$3$ is not a root of the equation $x^2 + 3x – 18 = 0.$

Answer

$3$ is a root of the equation $x^2 + 3x – 18 = 0.$

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Question 771 Mark

Write the negations of the following :Tirupati is in Andhra Pradesh.

Answer

The negations of the given statements are :
Tirupati is not in Andhra Pradesh.

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Question 781 Mark

Write the following compound statements symbolically:x is not irrational number but it is a square of an integer.

Answer

Let p : x is not irrational number
q : It is a square of an integer
Then the symbolic form of the given statement is p ∧ q
Note : If p : x is irrational number, then the symbolic form of the given statement is ~p ∧ q.

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Question 791 Mark

Write the following compound statements symbolically:Hima Das wins gold medal if and only if she runs fast.

Answer

Let p : Hima Das wins gold medal
q : She runs fast.
Then the symbolic form of the given statement is p ↔ q.

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Question 801 Mark

Write the following compound statements symbolically:

If ∆ ABC is right angled at B, then m∠A + m∠C = 90°.

Answer

Let p : ∆ ABC is right angled at B.
q : m∠A + m∠C = 90°.
Then the symbolic form of the given statement is p → q

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Question 811 Mark

Write the following compound statements symbolically:Angle is neither acute nor obtuse.

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Question 821 Mark

Write the following compound statements symbolically:

The angle is right angle if and only if it is of measure 90°.

Answer

Let p : The angle is right angle.
q : It is of measure 90°.
Then the symbolic form of the given statement is p↔q

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Question 831 Mark

Write the following compound statements symbolically:Triangle is equilateral or isosceles,

Answer

Let p : Triangle is equilateral.
q : Triangle is isosceles.
Then the symbolic form of the given statement is P∨q.

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Question 841 Mark

Write the following compound statements symbolically:Nagpur is in Maharashtra and Chennai is in Tamil Nadu.

Answer

Let p : Nagpur is in Maharashtra.
q : Chennai is in Tamil Nadu.
Then the symbolic form of the given statement is P∧q.

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Maths Answer the following questions in short.

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