Decide whether the set S = {−1, 0 ,1} is closed under each of the following operations. Be sure to justify each of your conclusions.

1. Multiplication
2. Addition
3. Subtraction
4. Division
5. Squaring
6. Square rooting
7. Averaging 2 numbers
8. Averaging 3 numbers

Answer is explained below. We can say that Subtraction is not closed under the set.

Given set S = {−1, 0 ,1}  we have to check the given set is closed or not.

1. Multiplication

Multiply the set S; the answer should also belong to the set S. For example, , 0 Є S and , -1 Є S, so the set S is closed under multiplication.

2. Addition

On adding the any two values of the set, the result belong to the set itself, for ex if we add Є S  but 1. so we can say that the given set is closed under addition.

3. Subtraction

The set S is not closed under subtraction because  -1 – 1  = -2, -2 is not the set element S.

4. Division

The set S is not belongs to closed division because when we divide one S element to another element, the answer does not belong to the set S, for example -1/0 = undefined.

5. Squaring

The set is closed under squaring,since we square an element of the set S, and the result also belongs to the set S. for example, and which belongs to the set S. Hence set S is closed.

Hence We can say that Subtraction is not closed under the set.

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Sol: 1. Multiplication

Multiply the set S; the answer should also belong to the set S. For example, -1 x 0 = 0 Є S and -1 x 1 = -1 Є S, Hence the set S is closed under multiplication.

2. Addition

When we add two sets S elements, the answer should also belong to the set S, for example, 1 + 0 = 1 Є S but 1 + 1 = 2, which does not belong to set S; hence S is not closed under addition.

3. Subtraction

The set S is not closed under subtraction because  -1 – 1  = -2, -2 is not the set S.

4. Division

The set S is not a closed division because when we divide one S element to another element, the answer does not belong to the set S, for example -1/0 = undefined.

5. Squaring

The set is closed under squaring if we square an element of the set S, and the result also belongs to the set S, for example, (0)2 = 0 and (-1)2 = 1, which belongs to the set S. Hence set S is closed under squaring.

6. Square rooting

The set S is not closed under square rooting because when we take the square root of an element of set S, the answer does not belong to the set S. √-1 = undefined.

7. Averaging two numbers

Averaging two numbers mean the sum of two numbers of set S is divided by two. For example, 1 + 0/2 = 1/2, which is not belongs to set S. hence set is not closed under averaging two numbers.

8. Averaging three numbers

The sum of three numbers of set S is divided by 3; the answer should belong to the set S.

-1 + 0 + 1 / 3= 0, 0 belongs to set. Hence the set S is closed under averaging three numbers.

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