The union of a subset and its complement

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The union of a subset S and its complement cS with respect to a set X is indeed equal to X. This can be proven by selecting any element from X and demonstrating that it belongs to either S or cS. The definitions of "complement" and "union" support this conclusion. The lack of discussion in most textbooks about such fundamental concepts raises questions about educational focus. Understanding these basic principles is essential for grasping more complex set theory topics.
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If S is a subset of X, and cS is the complement of S with respect to X, is the union of S and cS equal to X? Seems like a no-brainer but just want to be sure because I've yet to find a book that comments on this.
 
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Yes. You can pick any element of X and prove that it is in the union of S and cS. Use the definition of "complement" and "union"
 
Haha appropriate username, thanks. How come this isn't mentioned in most books? Seems like all the 'obvious' stuff is.
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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