Which of the Following Statements Are Correct?
Which of the following statements are correct? Write a correct form of each of the incorrect statements.
(i) \(a \subset \{a,b,c\}\)
(ii) \(\{a\} \in \{a,b,c\}\)
(iii) \(a \in \{\{a\},b\}\)
(iv) \(\{a\} \subset \{\{a\},b\}\)
(v) \(\{b,c\} \subset \{a,(b,c)\}\)
(vi) \(\{a,b\} \subset \{a,(b,c)\}\)
(vii) \(\Phi \in \{a,b\}\)
(viii) \(\Phi \subset \{a,b,c\}\)
(ix) \(\{x:x+3=3\}=\Phi\)
Solution
(i) Incorrect
Since \[ a \] is an element and not a set, the correct statement is \[ a \in \{a,b,c\} \]
(ii) Incorrect
The set \[ \{a\} \] is not an element of \[ \{a,b,c\} \]
Correct statement: \[ \{a\} \subset \{a,b,c\} \]
(iii) Incorrect
The elements of \[ \{\{a\},b\} \] are \[ \{a\} \] and \[ b \]
Correct statement: \[ \{a\} \in \{\{a\},b\} \]
(iv) True
Since every element of \[ \{a\} \] is contained in \[ \{\{a\},b\} \]
(v) Incorrect
The set \[ \{a,(b,c)\} \] contains the elements \[ a \] and \[ (b,c) \] but not \[ b \] and \[ c \] separately.
Correct statement: \[ (b,c) \in \{a,(b,c)\} \]
(vi) Incorrect
Since \[ b \notin \{a,(b,c)\} \]
Correct statement: \[ a \in \{a,(b,c)\} \]
(vii) Incorrect
The empty set is not an element of \[ \{a,b\} \]
Correct statement: \[ \Phi \subset \{a,b\} \]
(viii) True
The empty set is a subset of every set.
(ix) Incorrect
Solving \[ x+3=3 \] gives \[ x=0 \]
Therefore, \[ \{x:x+3=3\}=\{0\} \]