Question 6: Verify the properties for the sets \(A\), \(B\) and \(C\) given below:
\((a)\) \(\;\) \(A=\{1,2,3,4\}\), \(B=\{3,4,5,6,7,8\}\), \(C=\{5,6,7,9,10\}\)
\((i)\) Associative of Union
\((ii)\) Associativity of intersection
\((iii)\) Distributive of Union over intersection
\((iv)\) Distributive of intersection over union
Solution:
\((i)\) Associative of Union
\((A\cup B)\cup C=A\cup (B\cup C)\) is Associative of Union
\(\text{L.H.S.}=(A\cup B)\cup C\)
\(\ \ \ \ \ \ \ \ \ \ \ =[\{1,2,3,4\}\)\(\cup\)\(\{3,4,5,6,7,8\}]\)\(\cup\)\(\{5,6,7,9,10\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4,5,6,7,8\}\)\(\cup\)\(\{5,6,7,9,10\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4,5,6,7,8,9,10\}\)
\(\text{R.H.S.}=A\cup (B\cup C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4\}\)\(\cup\)\([\{3,4,5,6,7,8\}\)\(\cup\)\(\{5,6,7,9,10\}]\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4\}\)\(\cup\)\(\{3,4,5,6,7,8,9,10\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4,5,6,7,8,9,10\}\)
Hence, \(\text{L.H.S.}=\text{R.H.S.}\)
\((ii)\) Associativity of intersection
\((A\cap B)\cap C=A\cap (B\cap C)\) is Associative of Intersection
\(\text{L.H.S.}=(A\cap B)\cap C\)
\(\ \ \ \ \ \ \ \ \ \ \ =[\{1,2,3,4\}\)\(\cap\)\(\{3,4,5,6,7,8\}]\)\(\cap\)\(\{5,6,7,9,10\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{3,4\}\)\(\cap\)\(\{5,6,7,9,10\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)
\(\text{R.H.S.}=A\cap (B\cap C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4\}\)\(\cap\)\([\{3,4,5,6,7,8\}\)\(\cap\)\(\{5,6,7,9,10\}]\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4\}\)\(\cap\)\(\{5,6,7\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)
Hence, \(\text{L.H.S.}=\text{R.H.S.}\)
\((iii)\) Distributive of Union over intersection
\(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\) is Distributive of Union over intersection
\(\text{L.H.S.}=A \cup (B \cap C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4\}\)\(\cup\)\([\{3,4,5,6,7,8\}\)\(\cap\)\(\{5,6,7,9,10\}]\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4\}\)\(\cup\)\(\{5,6,7\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4,5,6,7\}\)
\(\text{R.H.S.}=(A \cup B) \cap (A \cup C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =[\{1,2,3,4\}\)\(\cup\)\(\{3,4,5,6,7,8\}]\)\(\cap\)\([\{1,2,3,4\}\)\(\cup\)\(\{5,6,7,9,10\}]\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4,5,6,7,8\}\)\(\cap\)\(\{1,2,3,4,5,6,7,9,10\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4,5,6,7\}\)
Hence, \(\text{L.H.S.}=\text{R.H.S.}\)
\((iv)\) Distributive of intersection over union
\(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\) is Distributive of intersection over union
\(\text{L.H.S.}=A \cap (B \cup C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4\}\)\(\cap\)\([\{3,4,5,6,7,8\}\)\(\cup\)\(\{5,6,7,9,10\}]\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{1,2,3,4\}\)\(\cap\)\(\{3,4,5,6,7,9,10\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{3,4\}\)
\(\text{R.H.S.}=(A \cap B) \cup (A \cap C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =[\{1,2,3,4\}\)\(\cap\)\(\{3,4,5,6,7,8\}]\)\(\cup\)\([\{1,2,3,4\}\)\(\cap\)\(\{5,6,7,9,10\}]\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{3,4\}\)\(\cup\)\(\{\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{3,4\}\)
Hence, \(\text{L.H.S.}=\text{R.H.S.}\)