In both cases, the statements on the left and right of the symbols share the same truth value. Yet they seem to be used and thought of differently. How is one different from the other?
In: Mathematics
Both symbols depend _a lot_ on context. From your text I guess this is about some logic setting, but even that is a pretty large field with multiple conventions all around.
So: context please?
≡ is a definition. It establishes what a shorter word/symbol/expression means. For example, you might say “x is positive ≡ x>0” to be precise about what you mean by “positive.” Definitions do not require proofs, but you must be careful not to abuse this fact. Simply using ≡ to establish some logical relationship that would be convenient for you is bad practice.
↔ (sometimes also called “if and only if” or “iff”) is a logical relationship that requires proof. For example, you might say that “x is positive ↔ x^3 is positive.” This is not true simply because you said it, but there is a sequence of logical statements (that includes the definition above!) that can show both that the left side implies the right side and that the right side implies the left side.
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