CAN ANY ONE THEORETICALLY PROVE THAT - * - IS +?????I actually noe the answer . i am just expecting some new methods. many people give logicall proofs. please give me theoritical proofs...
We are going to use the following definitons
(a) Additive identity property
0 is the additive identity and 0 + a = a + 0 = a
(b) Addive inverse property
For any number a there exists an additive inverse denoted by -a, which has the property that a + -a = -a + a = 0
(c) Multiplicative identity property
1 is the multiplicative identity and 1 * a = a * 1 = a
(d) Multiplicative property of 0
For any a 0 * a = a * 0 = 0
(d) Distributive property
For any three numbers a, b and c, a(b + c) = ab + ac
(e) Additive property of equality
For any a, b and c if a = b then a + c = b + c
(f) Transitive property
For any a, b, and c. If a = c and b = c then a = b.
Lemma : -a = (-1)a
a + -a = 0 Inverse property of addition
a* 0 = 0 Multiplicative property of zero
1 + -1 = 0 Inverse property of addition
a (1 + -1) = 0 because 0 = 1 + -1
a + (-1)a = 0 because of the distributive property
a + -a + (-1)a = -a because of the additive property of equality
0 + (-1)a = -a because of the inverse property of equality
(-1)a = -a because of the identity property of addition
b + -b = 0 Because of the definiton of additive inverse
-a * 0 = 0 By the multiplicative property of 0
-a (b + -b) = 0 By the transitive property
-a(b) + -a(-b) = 0 By the distributive property.
ab + -ab + (-a)(-b) = ab By the additive property of equality
0 + (-a)(-b) = ab By the additive inverse property.
(-a)(-b) = ab By the additive identity property