State Whether the Following Statements Are True or False
State whether the following statements are true or false:
(i) \(1 \in \{1,2,3\}\)
(ii) \(a \subset \{b,c,a\}\)
(iii) \((a) \in \{a,b,c\}\)
(iv) \(\{a,b\}=\{a,a,b,b,a\}\)
(v) The set \[ \{x:x+8=8\} \] is the null set.
Solution
(i) True
Since \[ 1 \] is an element of the set \[ \{1,2,3\} \]
(ii) False
The symbol \[ \subset \] is used for sets, but \[ a \] is an element, not a set.
(iii) False
The elements of the set are \[ a,b,c \] but \[ (a) \] is not considered a separate element here.
(iv) True
Repetition of elements does not change a set. Therefore, \[ \{a,a,b,b,a\}=\{a,b\} \]
(v) False
Solving \[ x+8=8 \] gives \[ x=0 \]
Hence, \[ \{x:x+8=8\}=\{0\} \] which is not a null set.