LaTeX Component-wise Inequality Symbols: $\preceq$ and $\succeq$ etc.

In reference¹, when talking about “polyhedron (polyhedra)”, the mathematical symbol $\preceq$ is used to denote component-wise inequality. Specifically, for two vectors $\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}}\in\mathbb{R}^n$,

• $\boldsymbol{\mathrm{x}}\preceq\boldsymbol{\mathrm{y}}$ denotes that each element of $\boldsymbol{\mathrm{x}}$ is less than or equal to the corresponding element in $\boldsymbol{\mathrm{y}}$.

and similarly we have:

• $\boldsymbol{\mathrm{x}}\succeq\boldsymbol{\mathrm{y}}$ … greater than or equal to …
• $\boldsymbol{\mathrm{x}}\prec\boldsymbol{\mathrm{y}}$ … less than…
• $\boldsymbol{\mathrm{x}}\succ\boldsymbol{\mathrm{y}}$ … greater than…

Sometimes, “$\le$”, “$\ge$”, “$<$”, and “$>$” can be used in place of above symbols to denote the same meanings, but the both sides of the symbol should be vectors, such as $\boldsymbol{\mathrm{x}}\le\boldsymbol{\mathrm{y}}$ and $\boldsymbol{\mathrm{x}}\ge\boldsymbol{\mathrm{y}}$.

In LaTeX, above component-wise inequality symbols can be typed using following commands, respectively²:

• \preceq (precedes or equals³): $\preceq$
• \succeq (succeeds or equals) : $\succeq$
• \prec (precedes): $\prec$
• \succ (succeeds): $\succ$

and other similar ones⁴:

• \nprec (does not precede): $\nprec$
• \npreceq ( neither precedes nor equals): $\npreceq$
• \precneqq: $\precneqq$
• \precnsim: $\precnsim$
• \precnapprox: $\precnapprox$
• \nsucc (does not succeed): $\nsucc$
• \nsucceq (neither succeeds nor equals): $\nsucceq$
• \succneqq: $\succneqq$
• \succnsim: $\succnsim$
• \succnapprox: $\succnapprox$

Of course, these symbols can be used in Markdown files as well.

Besides, sometimes the symbol $\preceq$ (resp. $\succeq$) also can be used to denote that the symmetric matrix $\boldsymbol{\mathrm{A}}$ is negative (resp. positive) semidefinite by $\boldsymbol{\mathrm{A}}\preceq0$ (resp. $\boldsymbol{\mathrm{A}}\succeq0$). See blog⁵.

References