Sarasvatī refers to the “Goddess of learning” and wife of Lord Brahmā. She usually sits on a white swan and holds a vīna (stringed instrument) in her hands.
The White Swan or the Rajahamsa, is known for its ability to separate milk from water. This symbolizes the power to discriminate between good and bad, right and wrong, the valuable and useless – basis of knowledge.
Goddess Saraswati or Vagdevi is the epitome of knowledge, speech and ultimate wisdom.
Hamsa, the white swan is bestowed with a special power. It can separate milk from water even though both are mixed and absorb milk leaving behind water only. This is called hamsa ksheera nyaya also. Thus, hamsa represents purity, perfection, supernatural and also liberation.
Hence, the Hamsa has come to be associated with Goddess Saraswati as her Vahana.