{"articles":{"set-equality@set-theory":{"content":"<p>An equivalence relation between sets. \\(A\\) and \\(B\\) are equal sets if they contain the same elements. \\(A = B\\) if when \\({x \\in{} A}\\), \\({x \\in{} B}\\) and when \\({x \\notin{} A}\\), \\({x \\notin{} B}\\). <br><br> Applying the contrapositive to the above, we get an equivalent definition: \\[{x \\in{} A \\iff{} x \\in{} B}\\] This definition can be recogized as a statement about subsets: \\[{A = B \\iff{} A \\subset{} B \\text{and} B \\subset{} A}\\]</p>","names":[[["Equality of sets",""]]]}},"style":"Notion"}