#Notation Issue

18 messages · Page 1 of 1 (latest)

outer zodiac
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Let X be a non-empty arbitrary set.
Let G ⊂ P(X) be an arbitrary family of subsets of X.
Is there any difference between f(G) and f*(G) defined in the picture or are they just two notations for the same set of Image of 𝐺.

image.png
inner rainBOT
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glass pike
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If f^{-1}(B) = A in G, then f(A) in f(G)

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so i think, f*(G) subset f(G)

glass pike
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I can't see the converse inclusion right now

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maybe it isn't there

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or maybe i made a mistake in the other inclusion as well

grand vortex
grand vortex
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take f : {-1, 1} -> R defined by x l-> x²

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and G = {{1}}

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See that A = {1} will verify:
f[A] = {1}
but:
f-1[{1}] = {-1, 1} which is NOT in G

glass pike
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right you are, thanks

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@outer zodiacin general these are different sets

outer zodiac
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Thank you @grand vortex
Thank you @glass pike

torpid nimbusBOT
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@outer zodiac has given 1 rep to @grand vortex @glass pike

sage bison
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+close