In mathematics, the word "all" typically refers to the entire set of elements within a particular context or scope. When you say "all" in a mathematical statement, you're indicating that the statement applies to every element in that set without exception.
For example:
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Universal Quantification: In logic and set theory, the statement "For all \( x \)" (often written as \( \forall x \)) means that a certain property or condition holds true for every element \( x \) in a specified set.
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Set Notation: If you say "all integers," you are referring to every integer in the set of integers, which includes ..., -3, -2, -1, 0, 1, 2, 3, ... .
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Functions and Domains: When discussing functions, you might say "the function is defined for all real numbers," meaning that the function accepts every real number as an input.
In general, "all" is used to indicate inclusivity and universality within mathematical reasoning and statements.