When ${ }^{19} \mathrm{Ne}$ emits a positron, it undergoes beta plus decay. In this process, a proton in the nucleus is transformed into a neutron, and a positron (a positively charged electron) is emitted, resulting in a change of the atomic number. So, ${ }^{19} \mathrm{Ne}$ becomes ${ }^{19} \mathrm{~F}$. The reaction can be represented as follows:

${ }_{10}^{19} \mathrm{Ne} \xrightarrow{-\mathrm{e}_{+1} \text { (positron) }}{ }_9^{19} \mathrm{~F}$

When ${ }^{14} \mathrm{C}$ emits a beta particle ( $\beta^{-}$particle), it undergoes beta minus decay, which is also known as electron emission. In this process, a neutron in the nucleus is transformed into a proton, and a beta minus particle or electron (denoted as $\beta^{-}$) is emitted, which increases its atomic number. So, ${ }^{14} \mathrm{C}$ becomes ${ }^{14} \mathrm{~N}$. The reaction can be represented as follows:

${ }_6^{14} \mathrm{C} \xrightarrow{-\mathrm{e}-1 \text { (beta particle) }}{ }_7^{14} \mathrm{~N}$