Question 7: Verify De Morgan’s Laws for the following sets:
\(U=\{1,2,3, \ldots, 20\}\), \(A=\{2,4,6, \ldots, 20\}\) and \(B=\{1,3,5, \ldots, 19\}\).
De Morgan’s Laws:
\( (i)\;\) \({(A\cup B)}’={A}’\cap {B}’ \)
\( (ii)\;\) \({(A\cap B)}’={A}’\cup {B}’ \)
\( (i)\;\) \({(A\cup B)}’={A}’\cap {B}’ \)
\(A\cup B=\{2,4,6, \ldots, 20\}\cup \{1,3,5, \ldots, 19\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,\ldots, 20\}\)
\(\text{L.H.S.}={(A\cup B)}’\)
\(\ \ \ \ \ \ \ \ \ \ \ =U-(A\cup B)\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,\ldots, 20\}\)\(\ -\ \)\(\{1,2,3,\ldots, 20\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)
\(\text{R.H.S.}={A}’\cap {B}’\)
\({A}’=U-A\)
\(\ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{1,2,3,\ldots, 20\}\)\(\ -\ \)\(\{2,4,6, \ldots, 20\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{1,3,5, \ldots, 19\}\)
\({B}’=U-B\)
\(\ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{1,2,3,\ldots, 20\}\)\(\ -\ \)\(\{1,3,5, \ldots, 19\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{2,4,6, \ldots, 20\}\)
\(\text{R.H.S.}={A}’\cap {B}’\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,3,5, \ldots, 19\}\)\(\cap\)\(\{2,4,6, \ldots, 20\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)
Therefore, \({(A\cup B)}’={A}’\cap {B}’ \)
\( (ii)\;\) \({(A\cap B)}’={A}’\cup {B}’ \)
\(A\cap B=\{2,4,6, \ldots, 20\}\cap \{1,3,5, \ldots, 19\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)
\(\text{L.H.S.}={(A\cap B)}’\)
\(\ \ \ \ \ \ \ \ \ \ \ =U-(A\cap B)\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,\ldots, 20\}\)\(\ -\ \)\(\{\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,\ldots, 20\}\)
\(\text{R.H.S.}={A}’\cup {B}’\)
\({A}’=U-A\)
\(\ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{1,2,3,\ldots, 20\}\)\(\ -\ \)\(\{2,4,6, \ldots, 20\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{1,3,5, \ldots, 19\}\)
\({B}’=U-B\)
\(\ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{1,2,3,\ldots, 20\}\)\(\ -\ \)\(\{1,3,5, \ldots, 19\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\ \)\(\{2,4,6, \ldots, 20\}\)
\(\text{R.H.S.}={A}’\cup {B}’\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,3,5, \ldots, 19\}\)\(\cup\)\(\{2,4,6, \ldots, 20\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,\ldots, 20\}\)
Therefore, \({(A\cap B)}’={A}’\cup {B}’ \)