Given A = {a, b, c, d} and * is defined as follows.

1) Commulative Property:
\(a * b=c\)  but  \(b \times a=d \Rightarrow a * b \neq b * a\)
\(a * c=b\) but \(c * a=c \Rightarrow a * c \neq c * a\) 
\(\therefore\) \(\text { * }\) is not commutative on A..
2) Associative Property: 
\((a+b) * c=c * c=a\) 
\(a *(b * c)=a * b=c\) 
\(\therefore(a * b) * c \neq a *(b * c)\) 
\(\text { * }\) is not associative on A.

(i) 4 + 7 = 12
it s a proposition as its truth value is F
(ii) What are you doing?
It is a question and not a proposition 
(iii) 3ⁿ ≤ 81, n ∈ N
It is a proposition as it is true when
n = 1, 2, 3, 4
(iv) Peacock is our national bird. It is a proposition as its truth value is T.
(v) How tall this mountain is!
This is an exclamation, not a proposition.

(i) If 6 + 2 = 5, then the milk is white.
Let p: 6 + 2 = 5 (F)
q: Milk is white (T)  
p ➝ q is having the truth value T
(ii) China is in Europe or \(\sqrt3\) is an integer.
p: China is in Europe (F)
q: \(\sqrt3\) is an integer (F)
p v q is having the truth value (F).
(iii) It is not true time 5 + 5 = 9 or Earth is a planet.
Let P: 5 + 5 = 9 is not true (T)
q: Earth is a planet (T
~p ∨ q is having the truth value T
(iv) 11 is a prime number and all the sides of a rectangle are equal.
p:11 is a prime number (T)
q: Allthe sides of arectangle areequal (F)
p _^ q is having the truth value F

Given m*n = mⁿ + n^m ∀m, n∈Z
Let us take - 2, 2∈Z
Consider m = -3, n = 2
(m * n) = (-2 * 2) = (-2)² + (2)^-2
\(=4+\frac { 1 }{ 4 } =\frac { 17 }{ 4 } \notin Z\)
∴ * is not a binary operation on Z

√b is not defined for negative values, b which also ∈ R.
Hence, a√b is not defined for all a, b ∈R
* is not a binary operation on R