Question

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How do the Vedas survive during the dissolution of the Universe?

Answer

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During the time of dissolution Lord Shiva doesn't destroys the Vedas because they enter the Panchakshari Mantra (Nama Shivaya) and thus Lord Shiva graces on them.

In the Linga Purana Chapter 85 "Glory of Panchakshara Mantra" Lord Shiva states to Godesses Parvati:

प्रलये समनुप्राप्ते नष्टे स्थावरजंगमे ।
नष्टे देवासुरे चैव नष्टे चोरगराक्षसे ।।
सर्वं प्रकृतिमापन्नं त्वया प्रलयमेष्यति ।
एकोहं संस्थितो देवी न द्वितियोस्ति कुत्रचित् ।।

During the time of dissolution, when all the mobile and immobile beings, the Gods and Asuras, Nagas and Rakshasas get destroyed and O godesses, all the things including you yourself get absorbed in the Prakriti then only I remain in the Universe. None else remain on earth.

Then, Lord Shiva states:

तस्मिन्वेदाश्च शास्त्राणि मंत्रे पञ्चाक्षरे स्थिताः ।
ते नाशं नैव संप्राप्ता मच्छक्त्या ह्यनुपालिताः ।।

At that point of time, the Vedas enter into the Panchakshari Mantra and they do not get destroyed because of my grace.

Thus Vedas survive themselves by entering in the Panchakshari Mantra and thus Lord Shiva graces on them.

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