Answer to Question #88778 in Discrete Mathematics for Shivam Nishad

Question #88778
et U be the set of positive integers 1, 2, 3, ...
etc., A be the set of odd positive integers and
B be the set of even positive integers. Verify
De Morgan's laws.
1
Expert's answer
2019-05-08T08:07:33-0400
"U=\\{x\\isin\\Z^+ \\}, A=\\{x\\isin\\Z^+|x\\ is\\ odd \\}, B=\\{x\\isin\\Z^+|x\\ is\\ even \\}"

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

"{A\\cup B}=\\{x\\isin\\Z^+|x\\ is\\ odd \\vee x\\ is\\ even \\}=\\{x\\isin\\Z^+ \\}=U"


"\\overline{A\\cup B}=\\overline{U}=\\{ \\}"

"\\overline{A}=\\{x\\isin\\Z^+|x\\ is\\ not\\ odd \\}=\\{x\\isin\\Z^+|x\\ is\\ even \\}=B"

"\\overline{B}=\\{x\\isin\\Z^+|x\\ is\\ not\\ even \\}=\\{x\\isin\\Z^+|x\\ is\\ odd \\}=A"

"\\overline{A}\\cap\\overline{B}=\\{x\\isin\\Z^+|x\\ is\\ even\\ \\wedge \\ x\\ is\\ odd\\}=\\{ \\}"

Hence


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



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

"A\\cap B=\\{x\\isin\\Z^+|x\\ is\\ odd\\ \\wedge \\ x\\ is\\ even\\}=\\{ \\}"

"\\overline{A\\cap B}=U-\\{ \\}=U"




"\\overline{A}=\\{x\\isin\\Z^+|x\\ is\\ not\\ odd \\}=\\{x\\isin\\Z^+|x\\ is\\ even \\}=B"

"\\overline{B}=\\{x\\isin\\Z^+|x\\ is\\ not\\ even \\}=\\{x\\isin\\Z^+|x\\ is\\ odd \\}=A"

"\\overline{A}\\cup\\overline{B}=B\\cup A=\\{x\\isin\\Z^+|x\\ is\\ even\\vee x\\ is\\ odd \\}=\\{x\\isin\\Z^+ \\}=U"

Hence


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


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