Answer to Question #289342 in Discrete Mathematics for Arjun

Question #289342



Show that if A, B, and C are sets, then A ∩ B ∩ C = A ∪ B ∪ C


•by showing each side is a subset of the other side.


•using a membership table.

1
Expert's answer
2022-01-24T15:03:18-0500

To show that "\\overline{A\\cap B\\cap C}=\\overline{A}\\cup \\overline{B}\\cup \\overline{C}"

Let "x \\in \\overline{A\\cup B\\cup C}"

"x \\in U \\text{ and } x \\notin A \\cap B \\cap C\\\\\nx \\in U \\text{ and } x \\notin A \\text{ or } x \\notin B \\text{ or } x \\notin C\\\\\nx \\in U \\text{ and } x \\notin A \\text{ or } x \\in U \\text{ and } x \\notin B \\text{ or } x \\in U \\text{ and } x \\notin C\\\\\nx\\in \\overline{A} \\text{ or } x\\in \\overline{B} \\text{ or } x\\in \\overline{C} \\\\\nx\\in \\overline{A} \\cup \\overline{B} \\cup \\overline{C}\\\\\n\\implies \\overline{A\\cup B\\cup C} \\subseteq \\overline{A} \\cup \\overline{B} \\cup \\overline{C}"

Conversely,

Let "x \\in \\overline{A} \\cup \\overline{B} \\cup \\overline{C}"

"x \\in \\overline{A} \\text{ or } x \\in \\overline{B} \\text{ or } x \\in \\overline{C} \\\\\nx \\in U \\text{ and } x \\notin A \\text{ or } x \\in U \\text{ and } x \\notin B \\text{ or } x \\in U \\text{ and } x \\notin C\\\\\nx \\in U \\text{ and } x \\notin A \\text{ or } x \\notin B \\text{ or } x \\notin C\\\\\nx \\in U \\text{ and } x \\notin A \\cap B \\cap C\\\\\nx \\in \\overline{A\\cup B\\cup C}\\\\\n\\implies \\overline{A} \\cup \\overline{B} \\cup \\overline{C}\\subseteq \\overline{A\\cup B\\cup C}"

Hence,


"\\overline{A\\cap B\\cap C}=\\overline{A}\\cup \\overline{B}\\cup \\overline{C}"

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