Question 6: Verify the properties for the sets \(A\), \(B\) and \(C\) given below:
\((c)\) \(\;\) \(A=N\), \(B=Z\), \(C=Q\)
\((i)\) 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
\(\text{L.H.S.}=(A\cup B)\cup C\)
\(\ \ \ \ \ \ \ \ \ \ \ =[N\)\(\cup\)\(Z]\)\(\cup\)\(Q\)
\(\ \ \ \ \ \ \ \ \ \ \ =Z\)\(\cup\)\(Q\)
\(\ \ \ \ \ \ \ \ \ \ \ =Q\)
\(\text{R.H.S.}=A\cup (B\cup C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =N\)\(\cup\)\([Z\)\(\cup\)\(Q]\)
\(\ \ \ \ \ \ \ \ \ \ \ =N\)\(\cup\)\(Q\)
\(\ \ \ \ \ \ \ \ \ \ \ =Q\)
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
\(\text{L.H.S.}=(A\cap B)\cap C\)
\(\ \ \ \ \ \ \ \ \ \ \ =[N\)\(\cap\)\(Z]\)\(\cap\)\(Q\)
\(\ \ \ \ \ \ \ \ \ \ \ =N\)\(\cap\)\(Z\)
\(\ \ \ \ \ \ \ \ \ \ \ =N\)
\(\text{R.H.S.}=A\cap (B\cap C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =N\)\(\cap\)\([Z\)\(\cap\)\(Q]\)
\(\ \ \ \ \ \ \ \ \ \ \ =N\)\(\cap\)\(Z\)
\(\ \ \ \ \ \ \ \ \ \ \ =N\)
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
\(\text{L.H.S.}=A \cup (B \cap C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =N\)\(\cup\)\([Z\)\(\cap\)\(Q\)
\(\ \ \ \ \ \ \ \ \ \ \ =\{\}\)\(\cup\)\(Z\)
\(\ \ \ \ \ \ \ \ \ \ \ =Z\)
\(\text{R.H.S.}=(A \cup B) \cap (A \cup C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =[N\)\(\cup\)\(Z]\)\(\cap\)\([N\)\(\cup\)\(Q]\)
\(\ \ \ \ \ \ \ \ \ \ \ =Z\)\(\cap\)\(Q\)
\(\ \ \ \ \ \ \ \ \ \ \ =Z\)
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
\(\text{L.H.S.}=A \cap (B \cup C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =N\)\(\cap\)\([Z\)\(\cup\)\(Q]\)
\(\ \ \ \ \ \ \ \ \ \ \ =N\)\(\cap\)\(Q\}\)
\(\ \ \ \ \ \ \ \ \ \ \ =N\)
\(\text{R.H.S.}=(A \cap B) \cup (A \cap C)\)
\(\ \ \ \ \ \ \ \ \ \ \ =[N\)\(\cap\)\(Z]\)\(\cup\)\([N\)\(\cap\)\(Q]\)
\(\ \ \ \ \ \ \ \ \ \ \ =N\)\(\cup\)\(N\)
\(\ \ \ \ \ \ \ \ \ \ \ =N\)
Hence, \(\text{L.H.S.}=\text{R.H.S.}\)
Explanation:
\(\text{Super Set}\)\(\ \cup \ \)\(\text{Sub Set}\)\(\ =\ \)\(\text{Super Set}\)
\(\text{Super Set}\)\(\ \cap \ \)\(\text{Sub Set}\)\(\ =\ \)\(\text{Sub Set}\)
\(Q\) is super set of \(Z\), and \(Z\) is super set of \(N\)