# Describing the set of elements that are in at least one of the sets A or B

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.

• ??.... 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. May 14 at 1:53
• 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. May 14 at 1:57
• @DougM Right, I agree. So am I misunderstanding the textbook problem? May 14 at 1:59
• @DougM So the textbook is accidentally answering the "exactly" question instead of the "at least" question which is the one they asked? May 14 at 2:01
• 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. May 14 at 4:02