Question

What is the bit string corresponding to the symmetric difference of two sets?

Short answer

Bit \(A = 0\) and \(B = 0\): Symmetric difference bit\( = 0\)

Bit\(A = 0\)and\(B = 1\): Symmetric difference bit\( = 1\)

Bit\(A = 1\)and\(B = 0\): Symmetric difference bit\( = 1\)

Bit \(A = 1\) and \(B = 1\): Symmetric difference bit\( = 0\)

Step-by-step solution

Explain symmetric difference

Symmetric difference \(A \oplus B\): all elements in \(A\) or in \(B\), but not in both.

If the\(ith\)bit in the string is a\(1\), then\(i\)is in the set.

If the \(ith\)bit in the string is a \(0\), then \(i\) is NOT in the set.

Explain symmetric difference on difference of two sets

If String \(A\) contains \(0\) on the \(ith\) bit and string \(B\) contains a \(0\) on the \(ith\) bit, then the symmetric difference \(A \oplus B\)contains a \(0\) on the \(ith\) bit.

If String \(A\) contains \(0\) on the \(ith\) bit and string \(B\) contains a \(1\) on the \(ith\) bit, then the symmetric difference \(A \oplus B\)contains a\(1\)on the\(ith\)bit.

If String \(A\) contains \(1\) on the \(ith\) bit and string \(B\) contains a \(0\) on the \(ith\) bit, then the symmetric difference \(A \oplus B\)contains a\(1\)on the\(ith\)bit.

If String \(A\) contains \(1\) on the \(ith\) bit and string \(B\) contains a \(1\) on the \(ith\) bit, then the symmetric difference \(A \oplus B\)contains a \(0\) on the \(ith\) bit.