1. The problem is to verify the set identity $B - A = A^c - B$ using a membership table. 2. Recall the definitions: - $B - A$ means elements in $B$ but not in $A$. - $A^c$ is the complement of $A$, i.e., elements not in $A$. - $A^c - B$ means elements in $A^c$ but not in $B$. 3. To verify the identity, we create a membership table listing all possible membership combinations for elements in sets $A$ and $B$. | Element | $A$ | $B$ | $B - A$ | $A^c$ | $A^c - B$ | |---------|-----|-----|---------|-------|-----------| | 1 | 0 | 0 | 0 | 1 | 1 | | 2 | 0 | 1 | 1 | 1 | 0 | | 3 | 1 | 0 | 0 | 0 | 0 | | 4 | 1 | 1 | 0 | 0 | 0 | 4. Explanation: - For element 1: Not in $A$ or $B$, so $B - A = 0$, $A^c = 1$, $A^c - B = 1$. - For element 2: In $B$ but not $A$, so $B - A = 1$, $A^c = 1$, $A^c - B = 0$. - For element 3: In $A$ but not $B$, so $B - A = 0$, $A^c = 0$, $A^c - B = 0$. - For element 4: In both $A$ and $B$, so $B - A = 0$, $A^c = 0$, $A^c - B = 0$. 5. Comparing columns $B - A$ and $A^c - B$, they are not equal for all elements. 6. Therefore, the identity $B - A = A^c - B$ is false. 7. The correct identity involving complements and differences is $B - A = B \cap A^c$. Final answer: The given identity is not true as shown by the membership table.