Question

Let A= {1, 2, {3, 4,}, 5}. Which of the following statements are incorrect and why?
(i) \[\left\{ {3,{\text{ }}4} \right\} \subset A\]

(ii) \[\left\{ {3,{\text{ }}4} \right\} \in A\]

(iii) \[\left\{ {\left\{ {3,{\text{ }}4} \right\}} \right\} \subset A\]

(iv) \[1 \in A\]

(v) \[1 \subset A\]

(vi) \[\left\{ {1,{\text{ }}2,{\text{ }}5} \right\} \subset A\]

(vii) \[\left\{ {1,{\text{ }}2,{\text{ }}5} \right\} \in A\]

(viii) \[\left\{ {1,{\text{ }}2,{\text{ }}3} \right\} \subset A\]

(ix) \[\phi \in A\]

(x) \[\phi \subset A\]

(xi) \[\left\{ \phi \right\} \subset A\]

Answer 1

Hint: Here in this problem we are given is to choose which ones are incorrect among the given options. Here we are going to use the concept subsets and element of a set and find out which one is correct and which was is not.

Complete step by step solution: The given set is \[A = \{ 1,2,\{ 3,4\} ,5\} .\]
Option (i)
It is given that \[\left\{ {3,4} \right\} \subset A\].
The given statement is incorrect because \[\{ 3,4\} \in A\], as it is an element of A and not a subset of A.
Option (ii)
It is given that \[\{ 3,4\} \in A\].
The given statement is correct because \[\{ 3,4\} \;\] is an element of A.
Option (iii)
It is given that \[\{ \{ 3,4\} \} \subset A.\]
The given statement is correct because \[\{ 3,4\} \in \{ \{ 3,4\} \} \] and \[\{ 3,4\} \in A\], so \[\{ \{ 3,4\} \} \subset A.\]
Option (iv)
It is given that \[1 \in A\].
The given statement is correct because 1 is an element of A.
Option (v)
It is given that \[1 \subset A\].
The given statement is incorrect because an element of a set can never be a subset of itself.
Option (vi)
It is given that \[\{ 1,2,5\} \subset A.\]
The given statement is correct because each element of \[\{ 1,2,5\} \] is also an element of A. Hence, \[\{ 1,2,5\} \] is a subset of A.
Option (vii)
It is given that \[\{ 1,2,5\} \in A.\]
The given statement is incorrect because \[\{ 1,2,5\} \;\] is not element of A.
Option (viii)
It is given that \[\{ 1,2,3\} \subset A\]
The given statement is incorrect because \[3 \in \{ 1,2,3\} ,\]but \[3 \notin A\].
Option (ix)
It is given that \[\phi \in A\].
The given statement is incorrect because \[\phi \in \{ \phi \} ,\] but \[\phi \notin A\].
Option (x)
It is given that \[\phi \subset A.\]
The given statement is correct because \[\phi \] is a subset of every set.
Option (xi)
It is given that \[\left\{ \phi \right\} \subset A\]
The given statement is incorrect as \[\left\{ \phi \right\}\] indicates the set containing \[\phi \], and as \[\phi \notin A\], hence \[\left\{ \phi \right\} \not\subset A\].

Hence, options (i), (v), (vii), (viii), (ix) and (xi) are incorrect.

Note: Most of the conditions can seem to be same. But you need to keep that in mind that every condition given here are different types of problem and to be treated differently.
Subset: A set A is a subset of a set B if every element of A is also an element of B.
When we say that \[x \in A\], that is x is one of the objects in the collection of (possibly many) objects in the set A.