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The example sentence from a Mathematics book:

0 is predecessor of 1 and 1 is successor of 0. Similarly 1 predecessor of 2 and 2 is successor of 1 and so on.

I thought that we can use" vice versa" as well:

0 is predecessor of 1 and 1 is successor of 0. Similarly 1 predecessor of 2 and 2 is successor of 1 and vice versa.

Can anyone please tell me whether their usage is different to each other, if not then which one is most formal than the other?

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No, "vice versa" means with the order changed, or with the relations reversed, or conversely. We use it to indicate that a statement would still be true if some of the words in the sentence (often the subject and object, it seems to me) were transposed.

At the moment, you may see that information at the Merriam Webster on-line entry, with an example: "...Alexandria Ocasio-Cortez... has endorsed Nixon (and vice versa)." The meaning is that if we switched the positions of the names to produce the claim that Nixon has endorsed Alexandria Ocasio-Cortez, that resulting claim would be true.

But if you should say that 1 is the predecessor of 2 and 2 is the successor of 1 and vice versa, you would be saying that 2 is the predecessor of 1 and 1 is the successor of 2. That that statement would be incorrect; because in fact 2 is not the predecessor of 1 and 1 is not the successor of 2.

The two expressions (and so on, vice versa) seem perfectly proper to me in formal or informal contexts. But I suppose that the words "and so on," and the expression "etc.", are often used by lazy speakers to hint at further examples in a way that leaves us just guessing what they might be. So perhaps one should think twice before using them.

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"Vice versa" is the shortened form of taking two things that are reversible and describing that reversible...ness. "...and so on" is a continuation where you're forcing the reader/listener to infer the next steps through logic.

However, technically "and so on" can include "vice versa" if the list that you're summing up with "and so on" contains sets of reversible things. For (an overly contrived) example:

All of my siblings have happy marriages: I love my wife and she loves me. My brother loves his wife and she loves him. My sister loves her husband and he loves her. My other sister loves her husband and he loves her.

...could be turned into...

Myself, my brother, and my two sisters all have happy marriages: I love my wife and vice versa. My brother loves his wife and vice versa, and so on.

This is not the greatest example, but it's the best one that I could come up with.

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No. Vice versa is used for swapping the order of 2 entities with similar relationship. so on is used for continuing a list of similar things. In your case, you are explaining the predecessor- successor relation between 2 consecutive numbers and after 2 such examples, your list continues. Hence "so on" is appropriate here. However, if the relation is similar from both ends, you can use vice versa e.g. 1 is neighbor to 2 and vice versa (meaning 1 & 2 share a similar relation to each other which is neighborhood and it means the same from both end, even when the positions are swapped)

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The two phrases so on and vice-versa mean different things and so are not interchangeable. So on indicates that you can keep generating examples or material -- often by following the example that was just provided (but not always). Vice-versa indicates an inversion. To illustrate vice versa, consider that the following sentences are equivalent:

  • The first baseman can throw to the second baseman, and the second baseman can throw to the first baseman.
  • The first baseman can throw to the second baseman, and vice-versa.

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