Is Max A $\le$ Sup B for A $\subseteq$ B?

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    Maximum Supremum
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SUMMARY

The discussion confirms that if set A is a subset of set B (A ⊆ B), then the supremum of A (sup A) is less than or equal to the supremum of B (sup B). It also raises the question of whether the maximum of A is defined, but establishes that the relationship between the supremums holds regardless of the maximum's existence. Thus, it is definitively true that max A ≤ sup B under the given conditions.

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  • Understanding of set theory and subset relations
  • Familiarity with the concepts of supremum and maximum in mathematical analysis
  • Basic knowledge of inequalities in real numbers
  • Ability to interpret mathematical notation and symbols
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  • Study the properties of supremum and infimum in real analysis
  • Explore the differences between maximum and supremum in various contexts
  • Learn about the implications of subset relations on bounded sets
  • Investigate examples of sets where max A is not defined
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for A$\subset$ B max A $\le$ sup B ? is it true ?
 
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Is the maximum defined for $A$? Regardless, it is true that if $A\subseteq B$, then $\sup A\le \sup B$.
 

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