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No. Let A = {a} (a singleton set) then P(A) = {a, 0} where 0 is the null (empty) set.
f(x) = x^{2} is a continuous function on the set R of real numbers, and (-1, 1) is an open set in R, but f(-1, 1) = [0, 1), and [0, 1) is not an open set in R. So, f is not an open function on R.
1. Null set or Empty set 2. Singleton set 3. Pair set
That refers to a set that has exactly one element. Also known as a "singleton".
The set of irrational numbers is NOT denoted by Q.Q denotes the set of rational numbers. The set of irrational numbers is not denoted by any particular letter but by R - Q where R is the set of real numbers.