Question

Justify if the following set is closed under the following binary operations: addition, subtraction, multiplication, and division. Let the set S = {all the integer solutions for x² – 1 = 0).

Answer 1

The set S = {1, -1} is not closed under addition or subtraction, but it is closed under multiplication and division. Closure under an operation means combining any two elements from the set using the operation should result in another element from the set.

Step-by-step explanation:

The question asks us to justify if the set S = {all the integer solutions for x² – 1 = 0} is closed under the binary operations of addition, subtraction, multiplication, and division. First, we find the integer solutions for the given equation, which are x = 1 and x = -1, as these values satisfy the equation, resulting in zero. This means that our set S consists of two elements: S = {1, -1}.

Now, we need to check if this set is closed under addition: combining any two elements from this set should yield an element that is also within the set. Through checking:

• 1 + 1 = 2 (Not in S)
• 1 + (-1) = 0 (Not in S)
• -1 + 1 = 0 (Not in S)
• -1 + (-1) = -2 (Not in S)

It is clear that S is not closed under addition.

For subtraction:

• 1 - 1 = 0 (Not in S)
• 1 - (-1) = 2 (Not in S)
• -1 - 1 = -2 (Not in S)
• -1 - (-1) = 0 (Not in S)

S is not closed under subtraction.

Considering multiplication:

• 1 × 1 = 1 (In S)
• 1 × (-1) = -1 (In S)
• -1 × 1 = -1 (In S)
• -1 × (-1) = 1 (In S)

S is closed under multiplication.

For division, division by zero is undefined, so we consider only non-zero combinations:

• 1 ÷ 1 = 1 (In S)
• 1 ÷ (-1) = -1 (In S)
• -1 ÷ 1 = -1 (In S)
• -1 ÷ (-1) = 1 (In S)

S is closed under division.