Question

Which of the following is true?
A. $a\in \{\{a\},b\}$
B. $\{b,c\}\subset \{a,\{b,c\}\}$
C. $\{a,b\}\subset \{a,\{b,c\}\}$
D. None of these.

Answer 1

Hint: To solve this question we will take each of the options given and then asses them according to the rules and properties of the sets and determine which among all the options given is correct.

Complete step by step solution:
 We are given four options about sets and we have to determine which of them is true.
We will check all the options and check which of them is true.
In the first option we are given $a\in \{\{a\},b\}$ which is not correct because it is mentioned that $a$ belongs element of set $\{\{a\},b\}$ but it should be written as $\left\{ a \right\}$. The curly brackets are missing hence $a$ is not the part of set $\{\{a\},b\}$ that’s why it is not the correct option.
The second option given is $\{b,c\}\subset \{a,\{b,c\}\}$which means that that $\{b,c\}$ is subset of the parent set $\{a,\{b,c\}\}$ which is true because $\{b,c\}$ is indeed subset of parent set $\{a,\{b,c\}\}$. Hence this is the correct option.
In the third option we are given $\{a,b\}\subset \{a,\{b,c\}\}$which means that $\{a,b\}$ is subset of the set $\{a,\{b,c\}\}$ but it is not correct because in the set $\{a,\{b,c\}\}$either $\left\{ a \right\}$or $\left\{ b,c \right\}$is a part of set and not $\{a,b\}$. Hence this option is also wrong.

Option ‘B’ is correct

Note: The symbol $\in $ which is read as “ belongs to” when used in set it means that element belongs to the particular set. For example, $\left\{ a \right\}\in \{\{a\},b\}$ it means that $a$ belongs element of set $\{\{a\},b\}$.
The symbol $\subset $ signifies subset which when used in a set it means that it is a part of the parent set. For example, in the option $\{b,c\}\subset \{a,\{b,c\}\}$, $\subset $ shows that $\{b,c\}$ is subset of set $\{a,\{b,c\}\}$.