Sort, SortBy, Ordering, OrderedQ

This small family of functions is essential to programming in Mathematica. Indeed, sorting has been central to computer programming and algorithm analysis since programming's inception in the late 1950s.

See posts for:

Sort, SortBy, Ordering, OrderedQ

OrderingQ

OrderingQ

OrderingQ is the predicate of the family, which by default returns True of False if a List (or List of arguments of any function) is in "canonical" order, defined as lowest numerical or symbolic value to highest.

In[269]:= numericalList = Range@12 // RandomSample[#, 12] &

Out[269]= {11, 2, 5, 9, 1, 3, 8, 7, 4, 12, 10, 6}

In[270]:= OrderedQ@numericalList

Out[270]= False

Sort by default sorts from low to high.

In[271]:= Sort@numericalList

Out[271]= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

In[272]:= OrderedQ@%

Out[272]= True

In[273]:= alphaList = CharacterRange["a", "z"] // RandomSample[#, 12] &

Out[273]= {"l", "d", "f", "x", "o", "z", "i", "y", "w", "s", "m", "g"}

In[274]:= OrderedQ@alphaList

Out[274]= False

Again, Sort by default sorts from low to high, technically using the ASCII code for the characters.

In[275]:= Sort@alphaList

Out[275]= {"d", "f", "g", "i", "l", "m", "o", "s", "w", "x", "y", "z"}

In[276]:= OrderedQ@%

Out[276]= True

In[277]:= ToCharacterCode@alphaList

Out[277]= {{108}, {100}, {102}, {120}, {111}, {122}, {105}, {121}, {119}, {115}, {109}, \
{103}}

In[278]:= Sort@%

Out[278]= {{100}, {102}, {103}, {105}, {108}, {109}, {111}, {115}, {119}, {120}, {121}, \
{122}}

We can give OrderedQ our own ordering function, as complex as we like, to decide if the List is ordered. In this example we simply override OrderedQ's default, which is normally in accord with Sort's of ordering from small to large, and ask OrderedQ if the List is ordered from large to small.

In[281]:= OrderedQ[Sort@numericalList, Greater]

Out[281]= False

[b]