Real Analysis Proofs
1 ♦ 0 = 0.a 0.a = 0.a 0.a = (0 + 0 ).a [ Since 0 is the additive identity ] 0.a = 0.a + 0.a [ Distributive Axiom ] -0.a + 0.a = -0.a + 0.a + 0.a [ Adding (-0.a) to both sides, the additive inverse of 0.a ] 0 = ( -0.a + 0.a ) + 0.a [ Associative Axiom ] 0 = 0 + 0.a 0 = 0.a [ Since 0 is the additive identity ] ♠ 2 ♦ -a = -1(a) (-a) = (-a) (-a) = (-a) + 0 [ Additive Identity ] (-a) = (-a) + 0.a [ Since 0 = 0.a (By above theorem (1)) ] (-a) = (-a) + ( 1 + (-1) ) a [ Additive Inverse of 1 ] (-a) = (-a) + ( 1.a + (-1)a ) [ Distributive Axiom ] (-a) = (-a) + ( a + (-1)a ) [ Multiplicative Identity ] (-a) = ( (-a) + a ) + (-1)a [ Associative Axiom ] (-a) = 0 + (-1)a [ Additive Identity ] (-a) = (-1)a [ Additive Identity ] ♠ 3 ♦ -(a + b) = (-a) + (-b) -(a + b) = -(a + b) –(a + b) = (-1)( a + b ) [ By above theorem (2) ] (-1)(a + b) = (-1)(a) + (-1)(b) [ Distributive Axiom ] -(a + b) = (-a) + (-b) [ By above theorem