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If A={1,2,3,4},B={3,4,5,6},U={1,2,3,4,5,6}A=\{1,2,3,4\},B=\{3,4,5,6\}, U=\{1,2,3,4,5,6\} then prove De Morgan's law. \newlineDe Morgan's law: first Law: (A∪B)′=A′∩B′(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime} Second Law: (A∩B)′=A′∪B′(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}

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Q. If A={1,2,3,4},B={3,4,5,6},U={1,2,3,4,5,6}A=\{1,2,3,4\},B=\{3,4,5,6\}, U=\{1,2,3,4,5,6\} then prove De Morgan's law. \newlineDe Morgan's law: first Law: (A∪B)′=A′∩B′(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime} Second Law: (A∩B)′=A′∪B′(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}
  1. Find Union of A and B: First, let's find the union of A and B. A∪B={1,2,3,4}∪{3,4,5,6}={1,2,3,4,5,6}A \cup B = \{1, 2, 3, 4\} \cup \{3, 4, 5, 6\} = \{1, 2, 3, 4, 5, 6\}
  2. Find Complement of Union in U: Now, let's find the complement of A∪BA \cup B in UU.(A∪B)′=U−(A∪B)={1,2,3,4,5,6}−{1,2,3,4,5,6}={}(A \cup B)' = U - (A \cup B) = \{1, 2, 3, 4, 5, 6\} - \{1, 2, 3, 4, 5, 6\} = \{\}
  3. Find Complements of A and B: Next, find the complements of A and B in U separately. \newlineA′=U−A={1,2,3,4,5,6}−{1,2,3,4}={5,6}A' = U - A = \{1, 2, 3, 4, 5, 6\} - \{1, 2, 3, 4\} = \{5, 6\}\newlineB′=U−B={1,2,3,4,5,6}−{3,4,5,6}={1,2}B' = U - B = \{1, 2, 3, 4, 5, 6\} - \{3, 4, 5, 6\} = \{1, 2\}
  4. Find Intersection of Complements: Now, let's find the intersection of A′A' and B′B'.A′∩B′={5,6}∩{1,2}={}A' \cap B' = \{5, 6\} \cap \{1, 2\} = \{\}

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