Meaning of $\{ a,b \}$, and comparison with $(a,b)$

Question

What does $\{a,b\}$ mean in real analysis?

I'm also little bit confused about set definition

Can you tell me the main difference between $(a,b)$ and $\{a,b\}$?

Thank you.

Answer 1

They are both sets, but of different nature. For instance,

$\{a, b\}$ is just a set two points, namely $a$ and $b$.

While $(a,b) = \{ x : a < x < b \} $ is the set of all point $x$ in the reals (since you mentioned real analysis) such that $x$ is between $a$ and $b$, so it contains a lot more points than $\{a, b\} $

$(a,b)$ can also be seen as an ordered pair. That is, a point on the plane $\mathbb{R}^2$. In this case $(a,b)$ is just a single point who lives in $\mathbb{R} \times \mathbb{R} $

Answer 2

The ordered pair $(a,b)$ may be defined as $(a,b)=\{a,\{a,b\}\}$.