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mgervasoni
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Thanks in advance. 1st day at calculus teacher wants proofs. They seem rudimentary but I've never done them and he doesn't help so I'm hoping someone here could please.
These are the axioms:
Addition:
For a, b, and c taken from the real numbers
A1: a+b is a real number also (closure)
A2: There exist 0, such that 0 + a = a for all a (existence of zero - an identity)
A3: For every a, there exist b (written -a), such that a + b = 0 (existence of an additive inverse)
A4: (a + b) + c = a + (b + c) (associativity of addition)
A5: a + b = b + a (commutativity of addition)
Multiplication:
For a, b, and c taken from the real numbers excluding zero
M1: ab is a real number also (closure)
M2: There exist an element, 1, such that 1a = a for all a (existence of one - an identity)
M3: For every a there exists a b such that ab = 1
M4: (ab)c = a(bc) (associativity of multiplication)
M5: ab = ba (commutativity of multiplication)
D1: a(b + c) = ab + ac (distributivity)
Prove these theorems using the Addition and Mult. Axioms:
Theorem #1: (-1) • a = -a
Theorem #2: if a • b = 0 and a ≠ 0, then b = 0
if a ≠ 0 and b ≠ 0, then a * b ≠ 0
Theorem #3: if a ≠ 0 and b ≠ 0 then 1/a •*1/b = 1/a•b
Here is the example he did in class:
prove a•0=0
line 1 a•0
line 2:A2: a•0+0
line 3:A3: a•0+{a•0+[-(a•0)]}
line 4:A4: {a•0+a•0}+[-(a•0)]
line 5:D1: a•(0+0)+[-(a•0)]
line 6:A2: a•0+[-(a•0)]
line 7:A3: 0
I've never done proofs this way so if anyone has any pointers or ways of thinking about it that would really help. Professor is no help. Thanks guys/girls.
These are the axioms:
Addition:
For a, b, and c taken from the real numbers
A1: a+b is a real number also (closure)
A2: There exist 0, such that 0 + a = a for all a (existence of zero - an identity)
A3: For every a, there exist b (written -a), such that a + b = 0 (existence of an additive inverse)
A4: (a + b) + c = a + (b + c) (associativity of addition)
A5: a + b = b + a (commutativity of addition)
Multiplication:
For a, b, and c taken from the real numbers excluding zero
M1: ab is a real number also (closure)
M2: There exist an element, 1, such that 1a = a for all a (existence of one - an identity)
M3: For every a there exists a b such that ab = 1
M4: (ab)c = a(bc) (associativity of multiplication)
M5: ab = ba (commutativity of multiplication)
D1: a(b + c) = ab + ac (distributivity)
Prove these theorems using the Addition and Mult. Axioms:
Theorem #1: (-1) • a = -a
Theorem #2: if a • b = 0 and a ≠ 0, then b = 0
if a ≠ 0 and b ≠ 0, then a * b ≠ 0
Theorem #3: if a ≠ 0 and b ≠ 0 then 1/a •*1/b = 1/a•b
Here is the example he did in class:
prove a•0=0
line 1 a•0
line 2:A2: a•0+0
line 3:A3: a•0+{a•0+[-(a•0)]}
line 4:A4: {a•0+a•0}+[-(a•0)]
line 5:D1: a•(0+0)+[-(a•0)]
line 6:A2: a•0+[-(a•0)]
line 7:A3: 0
I've never done proofs this way so if anyone has any pointers or ways of thinking about it that would really help. Professor is no help. Thanks guys/girls.