What are the differences between these two abbreviations? What are the appropriate situations to use each?
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As reported by the New Oxford American Dictionary:
E.g. is used in expressions similar to including, when you are not intending to list everything that is being discussed.
There's a neat way to remember which is which: "e.g." and "example" both start with the letter "e" (so read "e.g." as "for example"); and "i.e." and "in other words" both start with the letter "i".