An involution is a function whose inverse is itself. From the perspective of set theory and functions, if a relation is a function and is symmetric, then it is an involution.
- The function has the inverse , which is the same function, and thus is an involution.
- The logical NOT is an involution because .
- The additive negation is an involution because .
- The identity function is an involution because therefore, and . Hence, it is an involution.
- The multiplicative inverse is an involution because . In fact, for any is an involution.
- Function is an involution . This induces that both and are in f. By the definition of the inverse of a function, is the inverse of the function f. Therefore, the function f must contain . From this, it is obtained that . Simmilalry, we can show that . Thus, .
Rewriting the first line we have: function is an involution .
- A function is an involution iff it is symmetric about the line in the coordinate plane.
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