for the `beta^(+)` (positron) emission from a nucleus, there is another competing process known as electron capture. Electron from an inner orbit (say K shell) is captured by the nucleus and neutrino is emitted. Show that if `beta^(+)` emission is energetically allowed, electron capture is necessarily allowed but not vice -versa.

Explanation

To demonstrate that if β⁺ emission is energetically allowed, then electron capture is necessarily allowed, but not vice versa, we will analyze the energy equations for both processes.

### Step 1: Understanding β⁺ Emission
In β⁺ emission, a proton in the nucleus is converted into a neutron, emitting a positron (e⁺) and a neutrino (ν). The reaction can be represented as:
\[ X_{A}^{Z} \rightarrow Y_{A}^{Z-1} + e^{+} + \nu + Q_1 \]
where \( Q_1 \) is the energy released in this process.
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