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  1. PO
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    <p>f ∈ O(g) says, essentially</p> <blockquote> <p>For <strong>at least one</strong> choice of a constant <em>k</em> > 0, you can find a constant <em>a</em> such that the inequality 0 &lt;= f(x) &lt;= k g(x) holds for all x > a. </p> </blockquote> <p>Note that O(g) is the set of all functions for which this condition holds.</p> <p>f ∈ o(g) says, essentially</p> <blockquote> <p>For <strong>every</strong> choice of a constant <em>k</em> > 0, you can find a constant <em>a</em> such that the inequality 0 &lt;= f(x) &lt; k g(x) holds for all x > a.</p> </blockquote> <p>Once again, note that o(g) is a set.</p> <p>In Big-O, it is only necessary that you find a particular multiplier <em>k</em> for which the inequality holds beyond some minimum <em>x</em>. </p> <p>In Little-o, it must be that there is a minimum <em>x</em> after which the inequality holds no matter how small you make <em>k</em>, as long as it is not negative or zero.</p> <p>These both describe upper bounds, although somewhat counter-intuitively, Little-o is the stronger statement. There is a much larger gap between the growth rates of f and g if f ∈ o(g) than if f ∈ O(g). </p> <p>One illustration of the disparity is this: f ∈ O(f) is true, but f ∈ o(f) is false. Therefore, Big-O can be read as "f ∈ O(g) means that f's asymptotic growth is no faster than g's", whereas "f ∈ o(g) means that f's asymptotic growth is strictly slower than g's". It's like <code>&lt;=</code> versus <code>&lt;</code>.</p> <p>More specifically, if the value of g(x) is a constant multiple of the value of f(x), then f ∈ O(g) is true. This is why you can drop constants when working with big-O notation.</p> <p>However, for f ∈ o(g) to be true, then g must include a higher <em>power</em> of x in its formula, and so the relative separation between f(x) and g(x) must actually get larger as x gets larger.</p> <p>To use purely math examples (rather than referring to algorithms):</p> <p>The following are true for Big-O, but would not be true if you used little-o:</p> <ul> <li>x² ∈ O(x²) </li> <li>x² ∈ O(x² + x)</li> <li>x² ∈ O(200 * x²)</li> </ul> <p>The following are true for little-o:</p> <ul> <li>x² ∈ o(x³)</li> <li>x² ∈ o(x!)</li> <li>ln(x) ∈ o(x)</li> </ul> <p>Note that if f ∈ o(g), this implies f ∈ O(g). e.g. x² ∈ o(x³) so it is also true that x² ∈ O(x³), (again, think of O as <code>&lt;=</code> and o as <code>&lt;</code>)</p>
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    1. COYes-- the difference is in whether the two functions may be asymptotically the same. Intuitively I like to think of big-O meaning "grows no faster than" (i.e. grows at the same rate or slower) and little-o meaning "grows strictly slower than".
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    2. CO@Phil Good wording. I worked that into my answer.
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