■♥tr♦ t♦ ❈♦♥t❡♠♣♦r❛r② ▼❛t❤ ❯♥✐♦♥s ❛♥❞ ■♥t❡rs❡❝t✐♦♥s ♦❢ ■♥t❡r✈❛❧s ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❯❑
❆♥♥♦✉♥❝❡♠❡♥t ◮ ❨♦✉ ❤❛✈❡ ❛ ❤♦♠❡✇♦r❦ ❛ss✐❣♥♠❡♥t ❞✉❡ ♥❡①t ▼♦♥❞❛②✳ ◮ ▼✐♥✐✲❡①❛♠ ✷ ✐s ♥❡①t ❲❡❞♥❡s❞❛②✳
❯♥✐♦♥s ❛♥❞ ■♥t❡rs❡❝t✐♦♥s ♦❢ ■♥t❡r✈❛❧s ■♥t❡r✈❛❧ ❡✈❡♥ts ♦♥ t❤❡ r❡❛❧ ❧✐♥❡ ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ t♦ ❢♦r♠ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ❡✈❡♥ts ✉s✐♥❣ ✉♥✐♦♥s ❛♥❞ ✐♥t❡rs❡❝t✐♦♥s t♦ r❡♣r❡s❡♥t ♦✉t❝♦♠❡s ✐♥ ♦♥❡ ❡✈❡♥t ♦r t❤❡ ♦t❤❡r ✭✉♥✐♦♥s✮ ♦r ♦✉t❝♦♠❡s ✐♥ t❤❡ ♦✈❡r❧❛♣ ♦❢ ❜♦t❤ ❡✈❡♥ts ✭✐♥t❡rs❡❝t✐♦♥s✮✳
❯♥✐♦♥s ❛♥❞ ■♥t❡rs❡❝t✐♦♥s ♦❢ ■♥t❡r✈❛❧s ▲❡t Ω ❜❡ ❛♥ ✐♥t❡r✈❛❧ ♦❢ r❡❛❧ ♥✉♠❜❡rs✱ ❛♥❞ ❧❡t E ❛♥❞ F ❜❡ ❡✈❡♥t ✐♥t❡r✈❛❧s ✐♥ Ω ✳ ◮ ❆ r❡❛❧ ♥✉♠❜❡r ♦✉t❝♦♠❡ ✐s ✐♥ E ✐❢ ✐t ✐s ❜❡t✇❡❡♥ t❤❡ ❡♥❞♣♦✐♥ts ♦❢ E ✳ ◮ ❚❤❡ ❡✈❡♥t E � F ✐s t❤❡ ✉♥✐♦♥ ♦❢ E ❛♥❞ F ✳ ■t ✐s t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡r ♦✉t❝♦♠❡s ✐♥ E ♦r ✐♥ F ✭♦r ✐♥ ❜♦t❤✮✳ ◮ ❚❤❡ ✉♥✐♦♥ ❝♦✉❧❞ ❜❡ ❛ ❧❛r❣❡r ✐♥t❡r✈❛❧✱ ♦r t✇♦ s❡♣❛r❛t❡ ✐♥t❡r✈❛❧s✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡r E ❛♥❞ F ♦✈❡r❧❛♣✳ ◮ ❚❤❡ ❡✈❡♥t E � F ✐s t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ E ❛♥❞ F ✳ ■t ✐s t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡r ♦✉t❝♦♠❡s ✐♥ E ✱ ❛♥❞ ✐♥ F ✳ ◮ ■❢ E ❛♥❞ F ♦✈❡r❧❛♣✱ t❤❡♥ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ✐s t❤❡ ✐♥t❡r✈❛❧ ❢♦r♠❡❞ ❜② t❤❡ ♦✈❡r❧❛♣✳
❯♥✐♦♥ ♦❢ ❚✇♦ ❖✈❡r❧❛♣♣✐♥❣ ■♥t❡r✈❛❧s ▲❡t Ω = [ ✶✵ , ✶✼ ] , E = [ ✶✶ , ✶✺ ] ❛♥❞ F = [ ✶✸ , ✶✻ ] ✳ ❚❤❡ ❡✈❡♥t E � F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ♦r ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿ ◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✷ ✐s ✐♥ E � F ✱ ❜❡❝❛✉s❡ ✶✷ ✐s ✐♥ E ✭❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✮✳ ◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✺ . ✺ ✐s ✐♥ E � F ✱ ❜❡❝❛✉s❡ ✶✺ . ✺ ✐s ✐♥ F ✭❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✮✳ ◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✹ ✐s ✐♥ E � F ✱ ❜❡❝❛✉s❡ ✶✹ ✐s ✐♥ E ❛♥❞ ✐♥ F ✭❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠✮✳ ◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✼ ✐s ♥♦t ✐♥ E � F ✱ ❜❡❝❛✉s❡ ✶✼ ✐s ♥♦t ✐♥ E ♥♦r ✐♥ F ✳
❯♥✐♦♥ ♦❢ ❚✇♦ ❖✈❡r❧❛♣♣✐♥❣ ■♥t❡r✈❛❧s ▲❡t Ω = [ ✶✵ , ✶✼ ] , E = [ ✶✶ , ✶✺ ] ❛♥❞ F = [ ✶✸ , ✶✻ ] ✳ ❚❤❡ ❡✈❡♥t E � F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ♦r ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿
❯♥✐♦♥ ♦❢ ❚✇♦ ❖✈❡r❧❛♣♣✐♥❣ ■♥t❡r✈❛❧s ▲❡t Ω = [ ✶✵ , ✶✼ ] , E = [ ✶✶ , ✶✺ ] ❛♥❞ F = [ ✶✸ , ✶✻ ] ✳ ❚❤❡ ❡✈❡♥t E � F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ♦r ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿
❯♥✐♦♥ ♦❢ ❚✇♦ ❖✈❡r❧❛♣♣✐♥❣ ■♥t❡r✈❛❧s ▲❡t Ω = [ ✶✵ , ✶✼ ] , E = [ ✶✶ , ✶✺ ] ❛♥❞ F = [ ✶✸ , ✶✻ ] ✳ ❚❤❡ ❡✈❡♥t E � F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ♦r ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿ ❚❤✉s✱ E � F ✐s ❛❝t✉❛❧❧② t❤❡ ✐♥t❡r✈❛❧ [ ✶✶ , ✶✻ ] ✳ ■ts ❧❡♥❣t❤ ✐s ✶✻ − ✶✶ = ✺ . ❚❤✉s✱ = ✶✻ − ✶✶ ✶✼ − ✶✵ = ✺ F ) = Length of E � F � P ( E Length of Ω ✼
❯♥✐♦♥ ♦❢ ❚✇♦ ❙❡♣❛r❛t❡ ■♥t❡r✈❛❧s ▲❡t Ω = [ ✶✵ , ✶✼ ] , E = [ ✶✶ , ✶✸ ] ❛♥❞ F = [ ✶✹ , ✶✼ ] ✳ ❚❤❡ ❡✈❡♥t E � F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✸✱ ♦r ❜❡t✇❡❡♥ ✶✹ ❛♥❞ ✶✼✿ ◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✷ ✐s ✐♥ E � F ✱ ❜❡❝❛✉s❡ ✶✷ ✐s ✐♥ E ✳ ◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✺ . ✺ ✐s ✐♥ E � F ✱ ❜❡❝❛✉s❡ ✶✺ . ✺ ✐s ✐♥ F ✳ ◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✸ . ✺ ✐s ♥♦t ✐♥ E � F ✱ ❜❡❝❛✉s❡ ✶✸ . ✺ ✐s ♥♦t ✐♥ E ♥♦r ✐♥ F ✳
❯♥✐♦♥ ♦❢ ❚✇♦ ❙❡♣❛r❛t❡ ■♥t❡r✈❛❧s ▲❡t Ω = [ ✶✵ , ✶✼ ] , E = [ ✶✶ , ✶✸ ] ❛♥❞ F = [ ✶✹ , ✶✼ ] ✳ ❚❤❡ ❡✈❡♥t E � F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✸✱ ♦r ❜❡t✇❡❡♥ ✶✹ ❛♥❞ ✶✼✿
❯♥✐♦♥ ♦❢ ❚✇♦ ❙❡♣❛r❛t❡ ■♥t❡r✈❛❧s ▲❡t Ω = [ ✶✵ , ✶✼ ] , E = [ ✶✶ , ✶✸ ] ❛♥❞ F = [ ✶✹ , ✶✼ ] ✳ ❚❤❡ ❡✈❡♥t E � F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✸✱ ♦r ❜❡t✇❡❡♥ ✶✹ ❛♥❞ ✶✼✿
❯♥✐♦♥ ♦❢ ❚✇♦ ❙❡♣❛r❛t❡ ■♥t❡r✈❛❧s ▲❡t Ω = [ ✶✵ , ✶✼ ] , E = [ ✶✶ , ✶✸ ] ❛♥❞ F = [ ✶✹ , ✶✼ ] ✳ ❚❤❡ ❡✈❡♥t E � F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✸✱ ♦r ❜❡t✇❡❡♥ ✶✹ ❛♥❞ ✶✼✿ ❚❤✉s✱ E � F ✐s ❛❝t✉❛❧❧② t✇♦ ✐♥t❡r✈❛❧s✳ ■t ❤❛s ❛ t♦t❛❧ ❧❡♥❣t❤ ❢♦✉♥❞ ❜② ❛❞❞✐♥❣ ✭❝♦♠❜✐♥✐♥❣✮ t❤❡ ❧❡♥❣t❤s ♦❢ E ❛♥❞ F ✿ ▲❡♥❣t❤ ♦❢ E : ✸ − ✶ = ✷ ▲❡♥❣t❤ ♦❢ F : ✼ − ✹ = ✸ � ❚♦t❛❧ ❧❡♥❣t❤ ♦❢ E F : ✷ + ✸ = ✺ . ❍❡♥❝❡ = ✺ F ) = Total length of E � F � P ( E Length of Ω ✼ .
■♥t❡rs❡❝t✐♦♥ ♦❢ ❚✇♦ ■♥t❡r✈❛❧s ▲❡t Ω = [ ✶✵ , ✶✼ ] , E = [ ✶✶ , ✶✺ ] ❛♥❞ F = [ ✶✸ , ✶✻ ] ✳ ❚❤❡ ❡✈❡♥t E � F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ❛♥❞ ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿ ◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✷ ✐s ♥♦t ✐♥ E � F ✱ ❜❡❝❛✉s❡ ✶✷ ✐s ✐♥ E ✱ ❜✉t ♥♦t ✐♥ F ✳ ◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✺ . ✺ ✐s ♥♦t ✐♥ E � F ✱ ❜❡❝❛✉s❡ ✶✺ . ✺ ✐s ✐♥ F ✱ ❜✉t ♥♦t ✐♥ E ✳ ◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✹ ✐s ✐♥ E � F ✱ ❜❡❝❛✉s❡ ✶✹ ✐s ✐♥ E ❛♥❞ ✐♥ F ✭❜♦t❤ ♦❢ t❤❡♠✮✳
■♥t❡rs❡❝t✐♦♥ ♦❢ ❚✇♦ ■♥t❡r✈❛❧s ▲❡t Ω = [ ✶✵ , ✶✼ ] , E = [ ✶✶ , ✶✺ ] ❛♥❞ F = [ ✶✸ , ✶✻ ] ✳ ❚❤❡ ❡✈❡♥t E � F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ❛♥❞ ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿
■♥t❡rs❡❝t✐♦♥ ♦❢ ❚✇♦ ■♥t❡r✈❛❧s ▲❡t Ω = [ ✶✵ , ✶✼ ] , E = [ ✶✶ , ✶✺ ] ❛♥❞ F = [ ✶✸ , ✶✻ ] ✳ ❚❤❡ ❡✈❡♥t E � F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ❛♥❞ ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿
■♥t❡rs❡❝t✐♦♥ ♦❢ ❚✇♦ ■♥t❡r✈❛❧s ▲❡t Ω = [ ✶✵ , ✶✼ ] , E = [ ✶✶ , ✶✺ ] ❛♥❞ F = [ ✶✸ , ✶✻ ] ✳ ❚❤❡ ❡✈❡♥t E � F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ❛♥❞ ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿ ❚❤✉s✱ E � F ✐s ❛❝t✉❛❧❧② t❤❡ ✐♥t❡r✈❛❧ [ ✶✸ , ✶✺ ] ✳ ■ts ❧❡♥❣t❤ ✐s ✺ − ✸ = ✷ . ❍❡♥❝❡ F ) = Length of E � F = ✷ � P ( E ✼ . Length of Ω
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