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(i) Closure axiom : 

Let x, y ∈ G. Then x = a + b√2, y = c + d√2; a, b, c, d ∈ Q. 

x + y = (a + b√2) + (c + d√2) = (a + c) + (b + d) √2 ∈ G, 

Since (a + c) and (b + d) are rational numbers. 

∴ G is closed with respect to addition.

(ii) Associative axiom : Since the elements of G are all real numbers, addition is associative.

(iii) Identity axiom : There exists 0 = 0 + 0√2 ∈ G such that for all x = a + b√2 ∈ G. 

x + 0 = (a + b√2) + (0 + 0√2) 

= a + b√2 = x 

Similarly, we have 0 + x = x. 

∴ 0 is the identity element of G and satisfies the identity axiom.

(iv) Inverse axiom: For each x = a + b√2 ∈ G, there exists -x = (-a) + (-b) √2 ∈ G such that x + (-x) = (a + b√2) + ((-a) + (- b) √2) 

= (a + (-a)) + (b + (-b)) √2 = 0 

Similarly, we have (- x) + x = 0 . 

∴ (- a) + (-b) √2 is the inverse of a + b√2 and satisfies the inverse axiom.

(v) Commutative axiom: 

x + y = (a + c) + (b + d) √2 = (c + a) + (d + b) √2

= (c + d√2) + (a + b√2) 

= y + x, for all x, y ∈ G. 

∴ The commutative property is true. 

∴ (G, +) is an abelian group. Since G is infinite, we see that (G, +) is an infinite abelian group.

(i) Let a,b ∈ A (i.e.) a ≠ ±1 , b ≠ 1 

Now a * b = a + b – ab 

If a + b – ab = 1 ⇒ a + b – ab – 1 = 0 

(i.e.) a(1 – b) – 1(1 – b) = 0 

(a – 1)(1 – b) = 0 ⇒ a = 1, b = 1 

But a ≠ 1 , b ≠ 1 

So (a – 1) (1 – 6) ≠ 1 

(i.e.) a * b ∈ A. So * is a binary on A. 

To verify the commutative property:

Let a, b ∈ A (i.e.) a ≠ 1 , b ≠ 1 

Now a * b = a + b – ab and b * a = b + a – ba 

So a * b = b * a ⇒ * is commutative on A.

To verify the associative property: 

Let a, b, c ∈ A (i.e.) a, b, c ≠ 1 

To prove the associative property we have to prove that 

a * (b * c) = (a * b) * c

LHS: b * c = b + c – bc = D(say) 

So a * (b * c) = a * D = a + D – aD 

= a + (b + c – bc) – a(b + c – bc) 

= a + b + c – bc – ab – ac + abc 

= a + b + c – ab – bc – ac + abc … (1) 

RHS: (a * b) = a + b – ab = K(say) 

So (a * b) * c = K * c = K + c – Kc 

= (a + b – ab) + c – (a + b – ab) c 

= a + b – ab + c – ac – bc + abc 

= a + b + c – ab – bc – ac + abc ... (2) 

(ii) To verify the identity property: 

Let a ∈ A (a ≠ 1)

If possible let e ∈ A such that 

a * e = e * a = a 

To find e: 

a * e = a

(i.e.) a + e – ae = a

e(1 - a) = 0 

e = 0/(1 - a) = 0 (because a ≠ 1) 

So, e = (≠ 1) ∈ A 

(i.e.) Identity property is verified. 

To verify the inverse property: 

Let a ∈ A (i.e. a ≠ 1) 

If possible let a’ ∈ A such that 

To find a’: 

a * a’ = e 

(i.e.) a + a’ – aa’ = 0 

⇒ a'(1 – a) = – a

a' = -a/(1 - a) = a/(a - 1) A (because a ≠ 1)

So, a' ∈ A

⇒ For every a ∈ A there is an inverse a’ ∈ A such that

a* a’ = a’ * a = e 

⇒ Inverse property is verified.

(d) (¬p ∧ ¬q) → (¬p ∨ ¬q)

(i) Converse: If x and y are numbers such that x² = y² then x = y. 

Inverse: If x and y are numbers such that x ≠ y then x² ≠ y². 

Contrapositive : If x and v are numbers such that x² ≠ y² then x ≠ y. 

(ii) Converse: If a quadrilateral is a rectangle then it is a square. 

Inverse: If a quadrilateral is not a square then it is not a rectangle. 

Contrapositive : If a quadrilateral is not a rectangle then it is not a square

ASCII: Binary numbers are coded to represent characters in the computer memory. Several codes are used for this purpose. One most commonly used code is the American Standard Code for Information Interchange (ASCII). ASCII has been adopted by several American computer manufacturers as their computer’s internal code. This code is popular in data communications, is used almost exclusively to represent data internally in microcomputers, and is frequently found in the larger computers produced by some vendors.

ASCII is of two types:

ASCII-7 and ASCII-8. ASCII-7 is a 7-bit code that represents 128 (27) different characters.

ASCII-8 is an extended version of ASCII-7. It is an 8-bit code that represents 256 (28) different characters rather than 128. e.g. (i) A is given ASCII code 65.

Now if we convert 65 into Binary form we get 01000001 → 1 byte In the same way, every character has its own ASCII value after converting into binary code stored on the computer.

Postulates of Boolean Algebra: Boolean Algebra is an algebraic structure defined on a set of elements B together with two binary operators + and . provided the following postulates are satisfied:

(1) (a) Closure with respect to the operator +

(b) Closure with respect to the operator.

(2) (a) An identity element with respect to +, designated by 0 : X + 0 = 0 + X = X.

(b) An identify element with respect to designated by 1 : X . 1 = 1 . X = X.

(3) (a) Commutative with respect to + : X + Y = Y + X

(b) Commutative with respect to . : X . Y = Y . X

(4) (a) . is distributive over : : X . (Y + Z) = (X . Y) + (X . Z)

(b) + is distributive over . : X + (Y . Z) = (X + Y) . (X + Z)

(5) For every element X ∈ B, there exists an element \(\bar { X }\) ∈ B such that: (a) X × \(\bar { X }\)= 1

(b) X . \(\bar { X } \) = 0

The postulates listed above are called Huntington (1904) Postulates and need no proof. They are used to prove the theorems of Boolean Algebra.

p : 19 is a prime number 

q : All the angles of a triangle are equal

(i) ¬p ∧ q

(ii) p ∨ ¬ q

(iii) p ∧ q

(iv) ¬p

(a) 25

a * b = a + b – ab 

3 * (4 * 5) = 3 * (4 + 5 – 4(5)) 

= 3 * (9 – 20) 

= 3 * (-11) 

= 3 + (-11) – 3(-11) 

= 3 – 11 + 33 

= -8 + 33 = 25

(a) 4

The roots of fourth roots of unity are 1, -1, i, -i 

The identity element is 1 

(-i)⁴ = i⁴ = 1 

Order of (-i) = 4.

(i) p, → q is True : T

(ii) p ∨ q is False : F

(iii) ¬ p ∨ q is True : T

(iv) p ∧ q  is False : F

(d) If p and q are any two statements then p ⟷ q is a tautology.

(a) 8

The number of rows in truth table = 2ⁿ = 2³ = 8

(d) (ii), (iii), (iv)

Sentence (ii), (iii) and (iv) are statements 

(ii) Rose is a flower – True 

(iii) Milk is white – True 

(iv) 1 is a prime number 

∴ (ii), (iii), (iv) are statements 

(i) May god bless you. This statement can not be assigned True or False. 

∴ (i) is not a statements

(c) logically equivalent to p ∧ q