negationslash J then I J For any I \u03a9 there exists some x I Qsuch that x I I 3

# Negationslash j then i j for any i ω there exists

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negationslash = J then I J = . For any I Ω, there exists some x I Q such that x I I . 3 Define the function f : Ω −→ Q by f ( I ) = x I for any I Ω. Suppose I,J Ω and x I = x J . Then I J negationslash = . Therefore I = J . It follows that f is injective. Hence Ω lessorsimilar Q N . The set Ω is countable. (b) Let Ξ be a subset of P ( R ). Suppose the following conditions are satisfied: I is an open (non-empty) interval for any I Ξ. uniondisplay I Ξ I = R . Let Θ = braceleftbigg J P ( R ) : J = parenleftbigg x 1 n ,x + 1 n parenrightbigg for some x Q ,n N * bracerightbigg . For any x Q , there exists some n N * , I x Ξ such that parenleftbigg x 1 n x ,x + 1 n x parenrightbigg I x . (Note that x I x .) Define the function g : x −→ Ξ by g ( x ) = I x for any x Q . We have g ( Q ) lessorsimilar Q N . g ( Q ) is countable. We also have uniondisplay I g ( Q ) I = uniondisplay x Q I x uniondisplay x Q parenleftbigg x 1 n x ,x + 1 n x parenrightbigg = R . 4 Hence uniondisplay I g ( Q ) I = R . 2 The bijective function f : Map ( A, B × C ) −→ Map ( A, B ) × Map ( A, C ) is explicitly given by f ( ϕ ) = ( π B ϕ, π C ϕ ) for any ϕ Map ( A, B × C ). 3 We have applied the following result in analysis : For any s, t R , if s < t then there exists some q Q such that s < q < t . 4 We have applied the following result in analysis : For any r R , for any n N * , there exists q Q such that q 1 n < r < q + 1 n .

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• Spring '20
• Countable set, Basic concepts in set theory, Cardinal number

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