Question

Discuss the reflexive, symmetric, and transitive properties for asymptotic comparisons \((O, \Omega, \theta, o)\)

Short answer

The reflexivity, symmetry, and transitivity of the given notations are as follows: \n - \(O\) notation is reflexive and transitive, but not symmetric. \n - \(\Omega\) notation is reflexive and transitive, but not symmetric. \n - \(\theta\) notation is reflexive, symmetric, and transitive. \n - \(o\) notation is not reflexive, symmetric, or transitive.

Step-by-step solution

Refexivity

In the context of asymptotic notation, 'reflexivity' implies that a function will always be bounded by itself. \n For the given notations: \n - \(O\) notation is reflexive, because for any function \(f(n)\), \(f(n) = O(f(n))\). The function is certainly an upper bound for itself. \n - \(\Omega\) notation is reflexive, because for any function \(f(n)\), \(f(n) = \(\Omega\)(f(n))\). The function is an acceptable lower bound for itself. \n - \(\theta\) notation is reflexive as well, because for any function \(f(n)\), \(f(n) = \(\theta\)(f(n))\). The function is both an upper and lower bound for itself. \n - \(o\) notation is not reflexive, because for any \(f(n)\), \(f(n) \neq o(f(n))\). This is because \(o(f(n))\) implies that \(f(n)\) grows strictly slower than \(o(f(n))\). Therefore, a function cannot grow strictly slower than itself.

Symmetry

Syntactically, symmetry refers to the property of a binary relation where, if \(f(n)\) is related to \(g(n)\), then \(g(n)\) is related to \(f(n)\). \n For the given notations:\n - \(O\) notation is not symmetric since if \(f(n) = O(g(n))\), it does not imply \(g(n) = O(f(n))\).\n - \(\Omega\) notation is also not symmetric.\n - \(\theta\) notation is symmetric. If \(f(n) = \(\theta\)(g(n))\), then \(g(n) = \(\theta\)(f(n))\), by definition.\n - \(o\) notation is also not symmetric.

Transitivity

In terms of asymptotic notation, transitivity means that if \(f(n)\) is related to \(g(n)\) and \(g(n)\) is related to \(h(n)\), then \(f(n)\) is related to \(h(n)\). For the given notations:\n - \(O\) notation is transitive. If \(f(n) = O(g(n))\) and \(g(n) = O(h(n))\), then \(f(n) = O(h(n))\).\n - \(\Omega\) notation is transitive as well.\n - \(\theta\) notation is also transitive.\n - \(o\) notation, however, is not transitive. If \(f(n) = o(g(n))\) and \(g(n) = o(h(n))\), it does not necessarily imply that \(f(n) = o(h(n))\).

Key concepts

Reflexivity

Reflexivity in asymptotic notation means that a function can always be compared to itself in several important ways. This self-comparison is crucial in understanding how different asymptotic notations behave. Let's break it down for the various types of notations:

• Big O Notation (O): This notation is reflexive because any function \(f(n)\) will always satisfy \(f(n) = O(f(n))\). This essentially means that the function's growth rate is an upper bound for itself, reassuring us that no matter how a function behaves, it will always encapsulate its own growth limit.


• Omega Notation (\Omega): Reflexivity also holds for the \Omega notation, meaning \(f(n) = \Omega(f(n))\). Here, the function isn't just an upper bound for itself; it's a lower bound too, which is equally important in analyzing growth from the ground up.


• Theta Notation (\theta): This notation is reflexively clear because \(f(n) = \theta(f(n))\). It essentially states the function provides both the tightest possible upper and lower bounds for its growth.


• Small o Notation (o): The exception is the o notation, as \(f(n)\) cannot be equal to o(f(n))\. This is because the small o notation implies that the function's growth is strictly slower than what it is being compared to, a scenario that isn't possible with the function equating to itself.

Symmetry

Symmetry considers whether a relationship holds in both directions between two functions. Asymptotic notation provides interesting insights into this property:

• Big O and Omega Notations: Both O and \Omega lack symmetry. If \(f(n) = O(g(n))\), it simply means \(f(n)\) does not grow faster than \(g(n)\), not the other way around. Similarly, stating \(f(n) = \Omega(g(n))\) implies \(f(n)\) grows at least as fast as \(g(n)\), but not necessarily vice versa.


• Theta Notation: Symmetrically, if \(f(n) = \theta(g(n))\), then \(g(n) = \theta(f(n))\). It indicates a precise bounding behavior between two functions where they grow at exactly the same rate, suggesting an equivalence in terms of growth projection.


• Small o Notation: Due to its strictly less condition, the \(o\) notation does not support symmetry. If \(f(n) = o(g(n))\), it means that \(f(n)\) grows strictly slower than \(g(n)\) and vice versa cannot be inferred.

Transitivity

Transitivity explores how a relationship propagates through multiple functions. This is a vital aspect of understanding how different functions relate within asymptotic notations:

• Big O Notation: The transitive nature is evident here. If we have \(f(n) = O(g(n))\) and \(g(n) = O(h(n))\), we can conclude \(f(n) = O(h(n))\). Thus, O seamlessly connects these functions by their upper bound growth.


• Omega and Theta Notations: Transitivity is a key feature in these notations as well. For instance, \(f(n) = \Omega(g(n))\) and \(g(n) = \Omega(h(n))\) imply \(f(n) = \Omega(h(n))\). Similarly, if both \(f(n) = \theta(g(n))\) and \(g(n) = \theta(h(n))\) hold, then so does \(f(n) = \theta(h(n))\).


• Small o Notation: The exception arises here again, as transitivity does not apply to o\. Even when \(f(n) = o(g(n))\) and \(g(n) = o(h(n))\) are true, you can't definitively say \(f(n) = o(h(n))\), because each comparison only ensures slightly slower growth, which might not chain up overall.