Textbook problem: Use unions, intersections, and complements to express the set of elements that are in at least one of the sets $A$ or $B$, where $A$ and $B$ are subsets of the set $\Omega$.

My solution: Using juxtaposition to denote intersections my solution is $AB^c \, \cup \, BA^c \, \cup \, AB$ because I think it's the set of elements in $A$ but not $B$, in $B$ but not $A$, or in both $A$ and $B$.

Textbook solution: $(A\,\cup B)\,\cap\,(A\,\cap\,B)^c$ which I believe to be equivalent to $AB^c \, \cup \, BA^c$ using de Morgan's laws.

Question: Is the textbook solution a mistake or am I misunderstanding the problem?

Note that this is not for a course; I'm self-studying. This is an appendix problem on set notation and operations from an introduction to probability textbook. This particular appendix is preliminary material for the text proper.

  • 2
    $\begingroup$ ??.... The set of elements that are in at least one of A,B is $A\cup B$. But $(A\cup B)\cap (A\cap B)^c$ is the set of elements that are in A or in B but not in both A and B. $\endgroup$ May 14 at 1:53
  • 1
    $\begingroup$ The textbook answer gives the set of objects that is either in A or in B but not both. Exactly one of A or B. Your answer is equivalent to $A\cup B$ albeit in a more complicated formation. That is those objects in at least one of A and B. $\endgroup$
    – Doug M
    May 14 at 1:57
  • $\begingroup$ @DougM Right, I agree. So am I misunderstanding the textbook problem? $\endgroup$ May 14 at 1:59
  • $\begingroup$ @DougM So the textbook is accidentally answering the "exactly" question instead of the "at least" question which is the one they asked? $\endgroup$ May 14 at 2:01
  • 1
    $\begingroup$ If the textbook is really asking the question you say it is asking, and really giving the answer you say it is giving, it is time to find a better textbook. $\endgroup$ May 14 at 4:02


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