Question

If A is the subset of B and B is the subset of C then, prove that A is the subset of C.

Answer 1

Hint: We solve this problem by using the definition of a subset. A subset of a set is defined as a set such that every element that exists in the subset should also exist in the main set. That is if P is a subset of Q then every element of P exists in Q. By using this definition we prove that A is a subset of C based on given two conditions.

Complete step-by-step solution
We are given that A is the subset of B and B is the subset of C
We know that the definition of a subset is a set such that every element that exists in the subset should also exist in the main set. That is if P is a subset of Q then every element of P exists in Q.
By using the definition we can say that from A is a subset of B that is for every element of A exists in B.
Let us assume an element in A as
\[\Rightarrow x\in A\]
By using the definition we can say that
\[\Rightarrow x\in B\forall x\in A.........equation(i)\]
Now, let us take the second condition that is B is a subset of C
Here we have an element \['x'\] in B
So, we can write for the same element by using the definition of the subset that is
\[\Rightarrow x\in C\forall x\in B.........equation(ii)\]
Now, by combining both equation (i) and equation (ii) we get
\[\Rightarrow x\in C\forall x\in A\]
Here, we can see that all the elements in set A will also exist in set C.
By using the definition of subset we can conclude that A is a subset of C.
Hence the required result has been proved.

Note: We can explain the above question by using the examples.
We are given that A is the subset of B and B is the subset of C
Let us assume that set A has natural numbers.
Let us assume that set B as whole numbers.
Let us assume that the set C is Rational numbers.
Here, we can see that these examples satisfy the given condition that A is the subset of B and B is the subset of C that is
\[\begin{align}
  & \Rightarrow A\subset B \\
 & \Rightarrow B\subset C \\
\end{align}\]
Here, we can see that all the natural numbers that belong to set A also belong to the set of rational numbers that are set C.
So, we can write
\[\Rightarrow A\subset C\]
Therefore, we can conclude that A is a subset of C.
Hence the required result has been proved.