Question

Suppose that A, B, and C are sets. Prove or disprove that$(A-B)-C=(A-C)-B$

Short answer

$(A-B)-C=(A-C)-B$

Step-by-step solution

Step: 1

$$\begin{gathered} x\in (A-B)-C \\ \Rightarrow x\in (A-B)\wedge x\notin C \\ \Rightarrow x\in A\wedge x\notin B\wedge x\notin C \\ \Rightarrow x\in A\wedge (x\notin B\wedge x\notin C) \\ \Rightarrow x\in A\wedge (x\notin C\wedge x\notin B) \\ \Rightarrow (x\in A\wedge x\notin C)\wedge x\notin B \\ \Rightarrow x\in A-C\wedge x\notin B \\ \Rightarrow x\in (A-C)-B \\ (A-C)-B \end{gathered}$$

$(A-B)-C=(A-C)-B$