Show that the following four conditions are equivalent : 1. A B 2…

Question

Show that the following four conditions are equivalent :

$A \subset B$
$A – B = \phi$
$A \cup B = B$
$A \cap B = A$

✓

Answer

Here, first we will prove $(i) \Leftrightarrow (ii)$
Where, $(i) = A \subset B$ and $(ii) = A - B \neq\phi$
Suppose that $\mathrm{A} \subset \mathrm{B}$
Now, we need to prove $A-B \neq \phi$
If possible, let $A-B \neq \phi$
Thus, there exists $X \in A, X \neq B,$ but this is impossible as $A \subset B$
$\therefore A-B=\phi$
And $\mathrm{A} \subset \mathrm{B}\Rightarrow\mathrm{A}-\mathrm{B} \neq \phi$
Let suppose that $\mathrm{A}-\mathrm{B} \neq \phi$
Now, we have to prove: $\mathrm{A} \subset \mathrm{B}$
Let $\mathrm{X} \in \mathrm{A}$
It can be concluded that $\begin{equation} X \in B(\text { if } X \notin B, \text { then } A-B \neq \phi) \end{equation}$
Thus, $A - B = \begin{equation} \phi = A \subset B \end{equation}$
$\therefore (i) \Leftrightarrow (ii)$
Let us assume that $A \subset B$
To prove: $A \cup B = B$
$\begin{equation} \Rightarrow B \subset A \cup B \end{equation}$
Let us assume that, $\begin{equation} x \in A \cup B \end{equation}$
$\begin{equation} \Rightarrow x \in A \text { or } x \in B \end{equation}$
Taking case $I: \begin{equation} X \in B \end{equation}$
$\begin{equation} A \cup B=B \end{equation}$
Taking Case $II : \begin{equation} X \in A \end{equation}$
$\begin{equation} \Rightarrow X \in B(A \subset B) \end{equation}$
$\begin{equation} \Rightarrow A \cup B \subset B \end{equation}$
Let $A \cup B = B$
Let us assume that $\begin{equation} X \in A \end{equation}$
$\begin{equation} \Rightarrow X \in A \cup B(A \subset A \cup B) \end{equation}$
$\begin{equation} \Rightarrow X \in B(A \cup B=B) \end{equation}$
$\begin{equation} \therefore A \subset B \end{equation}$
Thus, $(i) \Leftrightarrow (iii)$
Now, to prove $(i) \Leftrightarrow (iv)$
Suppose that $A \subset B$
It can be observed that $\begin{equation} A \cap B \subset A \end{equation}$
Let $\begin{equation} X \in A \end{equation}$
To show: $\begin{equation} X \in A \cap B \end{equation}$
Since, $\begin{equation} A \subset B \text { and } X \in B \end{equation}$
Therefore, $\begin{equation} \mathrm{X} \in \mathrm{A} \cap \mathrm{B} \end{equation}$
$\begin{equation} \Rightarrow A \subset A \cap B \end{equation}$
$\begin{equation} \Rightarrow A=A \cap B \end{equation}$
Similarly, let us assume that $\begin{equation} \mathrm{A} \cap \mathrm{B}=\mathrm{A} \end{equation}$
Let $\begin{equation} X \in A \end{equation}$
$\begin{equation} \Rightarrow X \in A \cap B \end{equation}$
$\begin{equation} \Rightarrow X \in B \text { and } X \in A \end{equation}$
$\begin{equation} \Rightarrow A \subset B \end{equation}$
$\therefore (i) \Leftrightarrow (ii)$
Therefore, proved that $(i) \Leftrightarrow (ii) \Leftrightarrow (iii) \Leftrightarrow (iv)$

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