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

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Multiplication.

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

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.

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.

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

-1, 1 is a set of numbers that is closed under division. The rule is if you divide among you end up with a quotient that is in the set. 1/-1 or -1/1 = -1 (-1 is in the set)

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

1 No. 2 No. 3 Yes.

1 is the identity for multiplication. 1*x = x = x*1 for all rational x.

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