Question

Let $A=\{1,2,\{3,4\},5\}$. Which of the following statements are incorrect and why?
$\left(i\right)\{3,4\}\subset A\left(ii\right)\{3,4\}\in A$
$\left(iii\right)\left\{\{3,4\}\right\}\subset A\left(iv\right)1\in A$
$\left(v\right)1\subset A\left(vi\right)\{1,2,5\}\subset A$
$\left(vii\right)\{1,2,5\}\in A\left(viii\right)\{1,2,3\}\subset A$
$\left(ix\right)\varphi \in A\left(x\right)\varphi \subset A$
$\left(xi\right)\left\{\varphi \right\}\subset A$

Answer 1

Given: $A=\{1,2,\{3,4\},5\}$
$\left(i\right)\because 3\in \{3,4\}$ but $3\notin A$,
$\therefore \{3,4\}\subset A,$ is incorrect.

$\left(ii\right)\because \{3,4\}$ is an element of set $A$,
$\therefore \{3,4\}\in A$ is correct.

$\left(iii\right)\because \{3,4\}\in \left\{\{3,4\}\right\}$ and $\{3,4\}\in A$
$\therefore \left\{\{3,4\}\right\}\subset A$ is correct.

$\left(iv\right)1\in A$ is correct, because $1$ is an element of set $A$.

$\left(v\right)1\subset A$ is incorrect, because an element of a set can never be a subset of itself.

$\left(vi\right)\because$ Each element of {$1,2,5$} is also an element of set $A$,
$\therefore \{1,2,5\}\subset A$ is correct.

$\left(vii\right)\{1,2,5\}\in A$ is incorrect, because $\{1,2,5\}$ is not an element of set $A$.

$\left(viii\right)\{1,2,3\}\subset A$ is incorrect, because $3\in \{1,2,3\}$ but $3\notin A$.

$\left(ix\right)\varphi \in A$ is incorrect because $\varphi$ (empty set) is not an element of set $A$.

$\left(x\right)\varphi \subset A$ is correct because $\varphi$ (empty set) is a subset of every set.

$\left(xi\right)\left\{\varphi \right\}\subset A$ is incorrect because $\varphi \in \left\{\varphi \right\}$ but $\varphi \notin A$.