Q: Is the set -1 closed with respect to multiplication?

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Multiplication.

No. For a set to be closed with respect to an operation, the result of applying the operation to any elements of the set also must be in the set. The set of negative numbers is not closed under multiplication because, for example (-1)*(-2)=2. In that example, we multiplied two numbers that were in the set (negative numbers) and the product was not in the set (it is a positive number). On the other hand, the set of all negative numbers is closed under the operation of addition because the sum of any two negative numbers is a negatoive number.

Because the product of any two elements is also an element of the set.

of course!

No. Since -1 x -31 (= 31) would not be in the set.

No. 1 + 3 = 4, which is not odd. In fact, no pair of odds sums to an odd. So the set is not closed under addition.

Since that's a fairly small set, you should be able to check all combinations (for 2 numbers, there are only 4 possible multiplications), and see whether the result is in the set.

1 No. 2 No. 3 Yes.

The identity property for a set with the operation of multiplication defined on it is that the set contains a unique element, denoted by i, such that for every element x in the set, i * x = x = x * i The set need not consist of numbers, and the multiplication need not be the everyday kind of multiplication. Matrix multiplication is an example.

For any set of numbers, with the normal operation of multiplication defined on the set, there is only one identity, and that is 1.

When multiplication is defined over some domains, for every non-zero element X, in the domain, there exists a unique element Y, also in the domain such that X*Y = Y*X = 1 where 1 is the multiplicative identity. Such a value Y is written as X-1 or 1/X. Note that a multiplicative inverse need not exist. For example, the set of integers is closed under multiplication, but most elements do not have an inverse within the set.

1 is a whole number. It is the identity element with respect to multiplication but not addition.