On Upper and lower bounds - Wikipedia, the free encyclopedia
it says "2 and 5 are both lower bounds for the set { 5, 10, 34, 13934}"
how is 2 a lower bound when it is not even in the set?
thanks in advance
On Upper and lower bounds - Wikipedia, the free encyclopedia
it says "2 and 5 are both lower bounds for the set { 5, 10, 34, 13934}"
how is 2 a lower bound when it is not even in the set?
thanks in advance
A lower bound for a set $\displaystyle A\subset X$ is defined as number $\displaystyle x_0\in X$ with the property $\displaystyle x_0\leq a$ for all $\displaystyle a\in A$. It may be that $\displaystyle x_0$ in $\displaystyle A$, but it doesn't have to be. If $\displaystyle x_0\in A$ is a lower bound then it is automatically "the biggest lowerbound". Then $\displaystyle x_0$ is called a minimum of $\displaystyle A$.
If we look at your example, where we look at the given set as a subset $\displaystyle A\subset\mathbb{N}$. Then $\displaystyle x_0= 2, x_0= 5$ are both lower bounds. They satisfy $\displaystyle x_0\leq a$ for all $\displaystyle a\in A$. Even better $\displaystyle x_0 = 2$ is a minimum of $\displaystyle A$.