Step 1: Understand dimer formation in carboxylic acids.
Carboxylic acids such as propanoic acid form dimers due to intermolecular hydrogen bonding. Two molecules of acid associate through two hydrogen bonds.
\[
2RCOOH \rightleftharpoons (RCOOH)_2
\]
Step 2: Behaviour in vapour phase.
In the vapour phase, carboxylic acid molecules are not strongly solvated by solvent molecules. Therefore, they can associate with each other through hydrogen bonding and form dimers.
Hence, propanoic acid exists as dimer in vapour phase.
Step 3: Behaviour in aqueous solvent.
In aqueous medium, water molecules form strong hydrogen bonds with propanoic acid molecules. Water disrupts the acid-acid hydrogen bonding.
Therefore, dimer formation is not favoured in aqueous solvent.
Step 4: Behaviour in aprotic solvent.
Aprotic solvents do not donate hydrogen bonds effectively. Therefore, they do not break the hydrogen-bonded acid dimers easily.
Hence, propanoic acid can exist as dimer in aprotic solvents.
Step 5: Final conclusion.
Propanoic acid exists as dimer in vapour phase and aprotic solvent.
\[
\boxed{a,\ c}
\]