The Definition of “Less than or Equal to” or “Less than and Equal to” Depending on the Context

Abstract

In mathematics, the symbol “≤” is often interpreted as “less than or equal to”. However, behind this simple definition lies a deeper duality, which can subtly change into “less than and equal to” in a different situation. It shows fascinating properties, especially when applied to mathematical intervals. In this article, we will explore the complexity of the symbol “≤” in depth, analyze its different meaning in a variety of situations, and emphasize its unique role in standard inequality and interval mathematics. Furthermore, this deeper exploration will also strengthen our connection to the language of mathematic: when speakers say “less than or equal to” to express the symbol “≤”, such expression is literally insufficient and inaccurate, and “not greater than” or other proper expressions may be used instead. By the same token the two different definitions of the symbol “≥” can also be easily analyzed and concluded, and in similarity “not less than” or other proper expressions may be used to express the symbol “≥” instead of the current “greater than or equal to”.