Question

prove that if a and b are sets satisfying the property that a \ b = b \ a, then it must be the case that a = b

Answer 1

If
`\(a \backslash b = b \backslash a\)`, then
`\(a = b\)`.

Step-by-step explanation:

Let's start by understanding the given property
`\(a \backslash b = b \backslash a\)`. This implies that elements in
`\(a\)` but not in
`\(b\)` are the same as elements in
`\(b\) but not in \(a\)`. Mathematically, this can be expressed as
`\(a \cap \overline{b} = b \cap \overline{a}\)`, where
`\(\overline{b}\)` represents the complement of set
`\(b\).`

Now, let's analyze the implications of this equality. If
`\(a \cap \overline{b} = b \cap \overline{a}\)`, it means that the elements exclusive to
`\(a\)` are the same as the elements exclusive to
`\(b\)`. In other words, the sets
`\(a\) and \(b\)` have identical elements outside their intersection.

If
`\(a\)` and
`\(b\)` have the same elements outside their intersection, it implies that every element in
`\(a\)` is also in
`\(b\)` and vice versa. Therefore,
`\(a\)` and
`\(b\)` are equal, and we can conclude that if
`\(a \backslash b = b \backslash a\)`, then
`\(a = b\).`

In summary, the given property ensures that the elements exclusive to each set are the same, leading to the conclusion that the sets themselves are identical. This establishes the proof that if
`\(a \backslash b = b \backslash a\)`, then
`\(a = b\).`