由 Tzwan 於 星期五 七月 19, 2013 3:28 pm
令{y/x}為y/x的範圍, R: real numbers
{y/x} = R - (-3/5, 3/7)
proof:
claim: R - (-3/5, 3/7) ⊆ {y/x}
∀a ∈ R - (-3/5, 3/7),
取y=3, x=3a-1
∋ y/x = 3/3a-1 = a ∈ {y/x}
其中x的值範圍:
當|a|>1, 則|a-1| < 1, x ∈ [-3, 3] ⊆ [-5, 7]
當a ∈ [3/7, 1], 則a-1∈[7/3, 1], x ∈ [3, 7] ⊆ [-5, 7]
當a ∈ [-1, -3/5], 則a-1∈[-1, -5/3], x ∈ [-5, -3] ⊆ [-5, 7]
所以R - (-3/5, 3/7) ⊆ {y/x}
claim: {y/x} ⊆ R - (-3/5, 3/7)
{y/x}中最小正實數顯然是3/7
{y/x}中最大負實數顯然是-3/5
{y/x} ⊆ R
所以{y/x} ⊆ R - (-3/5, 3/7)