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No; here's a counterexample to show that the set of irrational numbers is NOT closed under subtraction:
pi - pi = 0.
pi is an irrational number. If you subtract it from itself, you get zero, which is a rational number. Closure would require that the difference(answer) be an irrational number as well, which it isn't. Therefore the set of Irrational Numbers is NOT closed under subtraction.
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Are rational numbers closed under subtraction?
Yes. They are closed under addition, subtraction, multiplication. The rational numbers WITHOUT ZERO are closed under division.
Is natural numbers a closed set under subtraction?
No.A set is closed under subtraction if when you subtract any two numbers in the set, the answer is always a member of the set.The natural numbers are 1,2,3,4, ... If you subtract 5 from 3 the answer is -2 which is not a natural number.
Is the sum of rational numbers always rational?
Yes. In general, the set of rational numbers is closed under addition, subtraction, and multiplication; and the set of rational numbers without zero is closed under division.
Are irrational numbers closed under division?
No. 4 root 2 and 2 root 2 are both irrational. Divide the first by the second you get 2. Which is not a member of the set of irrational numbers.
Are prime numbers closed under subtraction?
No.
