Beta Distribution

[Qui-Gon]

I am curious whether for [math](\alpha_1, \beta_1)\;, (\alpha_2, \beta_2) \in \mathbb{R}_{>0} \times \mathbb{R}_{>0}[/math] if [math]\text{Beta}(\alpha_1, \beta_1) = \text{Beta}(\alpha_2, \beta_2)[/math] then [math]\alpha_1=\alpha_2 \;\; \text{and} \;\; \beta_1=\beta_2 .[/math] In other words, distinct (alpha, beta) pairs generate distinct distributions. If this is not the case, meaning if you can find [math](\alpha_1, \beta_1) \neq (\alpha_2, \beta_2) \;\; \text{but} \;\; \text{Beta}(\alpha_1, \beta_1) = \text{Beta}(\alpha_2, \beta_2), [/math] is there a well-known functional relationship between such [math](\alpha_1, \beta_1)\;\; \text{and} \;\; (\alpha_2, \beta_2) \; ?[/math]
** Apologies for all the indents, could not figure out how to generate math-mode inline.