Question

Mention the closure property, associative law, commutative law, existance of identity, existance of inverse of each real number for each of the operations (i) addition (ii) multiplication on real numbers.

Answer 1

ADDITION PROPERTIES OF REAL NUMBERS

(i) Closure property: The sum of two real numbers is always a real number.
(ii) Associative law: (a + b) + c = a + (b + c) for all real numbers a, b and c.
(iii) Commutative law: a + b = b + a for all real numbers a and b.
(iv) Existence of additive identity: 0 is called the additive identity for real numbers.
As, for every real number a , 0 + a = a + 0 = a
(v) Existence of additive inverse: For each real number a, there exists a real number ($-$a) such that a + ($-$a) = 0 = ($-$a) + a. Here, a and ($-$a) are the additive inverse of each other.


MULTIPLICATION PROPERTIES OF REAL NUMBERS

(i) Closure property: The product of two real numbers is always a real number.
(ii) Associative law: (ab)c = a(bc) for all real numbers a, b and c.
(iii) Commutative law: a $\times$ b = b $\times$ a for all real numbers a and b.
(iv) Existence of multiplicative identity: 1 is called the multiplicative identity for real numbers.
As, for every real number a , 1 $\times$ a = a $\times$ 1 = a

(v) Existence of multiplicative inverse: For each real number a, there exists a real number $\left(\frac{1}{a}\right)$ such that a $\left(\frac{1}{a}\right)$ = 1 = $\left(\frac{1}{a}\right)$a. Here, a and $\left(\frac{1}{a}\right)$ are the multiplicative inverse of each other