Question

Using the axioms of an ordered field only, prove if x does not equal 0, then (1/x) does not equal 0.

Answer 1

To prove that if x does not equal 0, then (1/x) does not equal 0 using the axioms of an ordered field, assume x does not equal 0. Multiply both sides of the equation by x to get x * (1/x) = 0 * x. The left side simplifies to 1, since x * (1/x) is the multiplicative inverse of x. The right side simplifies to 0 since anything multiplied by 0 is 0. Therefore, 1 = 0, which is a contradiction. Since the assumption led to a contradiction, our initial statement that x does not equal 0 must be true. So, if x does not equal 0, then (1/x) does not equal 0.

Step-by-step explanation:

To prove that if x does not equal 0, then (1/x) does not equal 0 using the axioms of an ordered field:

1. Assume x does not equal 0.
2. Multiply both sides of the equation by x to get x * (1/x) = 0 * x.
3. The left side simplifies to 1, since x * (1/x) is the multiplicative inverse of x.
4. The right side simplifies to 0 since anything multiplied by 0 is 0.
5. Therefore, 1 = 0, which is a contradiction.
6. Since the assumption led to a contradiction, our initial statement that x does not equal 0 must be true.
7. So, if x does not equal 0, then (1/x) does not equal 0.