The conclusion in mathematics generally refers to the full prosecution of the steps that providing the solution for the problems in general. Some problems consists of several conditions that are to be checked for the exact solution that eventually occurs in the problems. This is the general way of producing the conclusions in mathematics.

- Show that (R − {0}, .) is an infinite abelian group. Here ‘.’ denotes usual multiplication.

**Solution: ** (i) * Closure axiom :*Since product of two non-zero real numbers is again a non-zero a real number. i.e., ∀ a, b ∈ R, a . b ∈
R.

(ii) **Associative axiom :** Multiplication is always associative in R− {0} i.e., a . (b . c) = (a . b) . c ∀ a, b, c ∈ R −
{0} ∴ associative axiom is
true.

(iii) **Identity axiom :** The identity element is 1 ∈ R − {0} under multiplication and 1 . a = a . 1 = a, ∀ a ∈ R − {0}

∴ Identity axiom is true.

(iv) **Inverse axiom :** ∀ a ∈ R − {0}, 1/a ∈ R − {0} such that

a .1/a =1/a . a = 1 (identity element). ∴ Inverse axiom is true. ∴ (R − {0}, .) is a group.

(v) ∀ a, b ∈ R − {0}, a . b = b . a

∴ Commutative property is true. ∴ (R − {0}, .) is an abelian group.

(vi) Further R − {0} is an infinite set, (R − {0}, .) is an infinite abelian

group.

**Conclusion means: ** In this problem the given set is proved as an infinite abelian group by satisfying the closure axiom, associative axiom, identity axiom,
inverse axiom, commutative property, infinite abelian forms

This is the general way of producing the conclusions in mathematics.

**Pro : ** A Tractor contains 16 red tank, 15 green tank, 18 blue tank and 13 yellow tank. If a tank is, choose at random from the tank, what is the probability of
choosing a red tank from the Tractor?

**Sol :** Given that, 14 colors are red, green, blue and yellow

Probability = n / N

P (total number of event) N = 16 + 15 + 18 + 13 = 62

P (probability of red) = 16

Probability = n / N

= 16 / 62

= 4 / 13

Solution is = 4 / 13

The above step provides the complete conclusion meaning for the above problem.