# Question on intervals

• Jan 24th 2013, 02:25 PM
sakuraxkisu
Question on intervals
My question is:

i) Is the following statement true or false, and explain why: "If a is less than or equal to b, then (-infinity, a) is a subset of (-infinity, b)".

I drew a diagram which showed that this was true, but how would you go about proving it formally?

Any help would be appreciated :)
• Jan 24th 2013, 03:41 PM
jakncoke
Re: Question on intervals
Assume $\displaystyle a \leq b$ then if a number $\displaystyle p \in (-\infty, a)$ then $\displaystyle p < a$, so $\displaystyle p < a \leq b$ thus $\displaystyle p < b$ so $\displaystyle p \in (-\infty, b)$
• Jan 24th 2013, 04:57 PM
HallsofIvy
Re: Question on intervals
The standard way to prove "A is a subset of B" is to start "if $\displaystyle x\in A$, then use the definitions of A and B to conclude "$\displaystyle x\in B$".

To prove "If a is less than or equal to b, then (-infinity, a) is a subset of (-infinity, b)":
If $\displaystyle p\in (-\infty, a)$ then p< a. Since $\displaystyle a\le b$ and "<" is transitive, $\displaystyle p<a\le b$ so $\displaystyle p\in (-\infty, b)$.