Question

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) For all sets \(A\) and \(B\), if \(A-B=\varnothing\), then \(B \neq \varnothing\).

Short answer

The statement is false. A counterexample is when both \(A\) and \(B\) are empty sets. In this particular case, \(A - B = \varnothing\) is true, but \(B \neq \varnothing\) is false.

Step-by-step solution

Understand Set Difference

The difference of two sets, \(A \)and \(B\), denoted as \(A - B\), or \(A \setminus B\), is the set of elements that are in \(A\) but not in \(B\). If this set difference is an empty set, \(\varnothing\), it signifies that all elements of \(A\) are also in \(B\). It does not necessarily indicate that set \(B\) cannot be an empty set.

Consider the Condition where \(A = \varnothing\)

While most of the times, the difference \(A - B = \varnothing\) would mean that set \(B\) is non-empty, there is an exception. When \(A\) itself is an empty set, then \(A - B = \varnothing\) is true regardless of whether \(B\) is empty or non-empty. For, there are no elements in set \(A\) to be subtracted.

Disprove the Statement

To disprove the statement, just provide a counterexample. If you let set \(A = \varnothing\) and set \(B = \varnothing\), then \(A - B = \varnothing\) is true, but \(B \neq \varnothing\) is false. Thus, the statement 'For all sets \(A\) and \(B\), if \(A - B = \varnothing\), then \(B \neq \varnothing\)' is incorrect, as it does not hold true for all conceivable scenarios.