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

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

Explanation

Solution

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
⇒x∈A\Rightarrow x\in A
By using the definition we can say that
⇒x∈B∀x∈A.........equation(i)\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′'x' in B
So, we can write for the same element by using the definition of the subset that is
⇒x∈C∀x∈B.........equation(ii)\Rightarrow x\in C\forall x\in B.........equation(ii)
Now, by combining both equation (i) and equation (ii) we get
⇒x∈C∀x∈A\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

& \Rightarrow A\subset B \\\ & \Rightarrow B\subset C \\\ \end{aligned}$$ Here, we can see that all the natural numbers that belong to set A also belong to the set of rational numbers that are set C. So, we can write $$\Rightarrow A\subset C$$ Therefore, we can conclude that A is a subset of C. Hence the required result has been proved.