Let \(f:X\to Y\) be a function from one set \(X\) to another set \(Y\), let \(S\) be a subset of \(X\), and let \(U\) be a subset of \(Y\). What, in general, can one say about \(f^{-1}(f(S))\) and \(S\)?. What about \(f(f^{-1}(U))\) and \(U\)?
- \(\bullet\) Show that \(f^{-1}(f(S))\subset S\)
- \(\Vdash\) \(f^{-1}(f(S))\subset S\)
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\(\equiv\) { Definition 3.4.1 }
- \(f^{-1}(\{f(x):x\in S\})\subset S\)
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\(\equiv\) { Definition 3.4.4 }
- \(\{x\in X:f(x)\in \{f(x):x\in S\}\}\subset S\)
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\(\equiv\) { Definition 3.1.15 }
- \(\text{True}\)
- \(\square\)
- \(\bullet\) Show that \(f(f^{-1}(U))\subset U\)
- \(\Vdash\) \(f(f^{-1}(U))\subset U\)
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\(\equiv\) { Definition 3.4.4 }
- \(f(\{x\in X: f(x)\in U\})\subset U\)
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\(\equiv\) { Definition 3.4.1 }
- \(\{f(x):x\in\{x\in X: f(x)\in U\}\} \subset U\)
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\(\equiv\) { Definition 3.1.15 }
- \(\text{True}\)
- \(\square\)