Let + (addition) be a binary operation on the set of odd numbers S. The set S is closed under + if for all a, b ϵ S, we also have a + b ϵ S.
Let 3, 5 ϵ the set of odd numbers
3 + 5 = 8 (8 is not an odd number)
Since 3 + 5 = 8 is not an element of the set of the odd numbers, the set of the odd numbers is not closed under addition.
Rational numbers are closed under addition, subtraction, multiplication. They are not closed under division, since you can't divide by zero. However, rational numbers excluding the zero are closed under division.
If you mean the set of non-negative integers ("whole numbers" is a bit ambiguous in this sense), it is closed under addition and multiplication. If you mean "integers", the set is closed under addition, subtraction, multiplication.
Yes. The empty set is closed under the two operations.
They form a closed set under addition, subtraction or multiplication.
No.
The numbers are not closed under addition because whole numbers, even integers, and natural numbers are closed.
Rational numbers are closed under addition, subtraction, multiplication. They are not closed under division, since you can't divide by zero. However, rational numbers excluding the zero are closed under division.
Yes. The set of real numbers is closed under addition, subtraction, multiplication. The set of real numbers without zero is closed under division.
If you mean the set of non-negative integers ("whole numbers" is a bit ambiguous in this sense), it is closed under addition and multiplication. If you mean "integers", the set is closed under addition, subtraction, multiplication.
Yes. The empty set is closed under the two operations.
Yes. They are closed under addition, subtraction, multiplication. The rational numbers WITHOUT ZERO are closed under division.
The set of rational numbers is closed under all 4 basic operations.
No. The set of rational numbers is closed under addition (and multiplication).
They form a closed set under addition, subtraction or multiplication.
They are closed under all except that division by zero is not defined.
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.
The set of even numbers is closed under addition, the set of odd numbers is not.