Equals Sign
The statement "f(x) is O(g(x))" as defined above is usually written as f(x) = O(g(x)). Some consider this to be an abuse of notation, since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have. As de Bruijn says, O(x) = O(x2) is true but O(x2) = O(x) is not. Knuth describes such statements as "one-way equalities", since if the sides could be reversed, "we could deduce ridiculous things like n = n2 from the identities n = O(n2) and n2 = O(n2)." For these reasons, it would be more precise to use set notation and write f(x) ∈ O(g(x)), thinking of O(g(x)) as the class of all functions h(x) such that |h(x)| ≤ C|g(x)| for some constant C. However, the use of the equals sign is customary. Knuth pointed out that "mathematicians customarily use the = sign as they use the word 'is' in English: Aristotle is a man, but a man isn't necessarily Aristotle."
Read more about this topic: Big O Notation, Matters of Notation
Famous quotes containing the words equals and/or sign:
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—J. August Strindberg (1849–1912)
“If we define a sign as an exact reference, it must include symbol because a symbol is an exact reference too. The difference seems to be that a sign is an exact reference to something definite and a symbol an exact reference to something indefinite.”
—William York Tindall (1903–1981)