Question

Suppose that A, B and C are sets such that \(A \subseteq B\) and \(B \subseteq C\). Show that \(A \subseteq C\).

Short answer

It is shown that A is a subset of C or\(A \subseteq C\).

Step-by-step solution

Given Data:

If every element of X is also an element of Y, then X is a subset of Y considering X and Y are not the same set.

It is given that:

\(A \subseteq B\)and \(B \subseteq C\)

To show that \(A \subseteq C\)

Let \(A \subset B\), \(x \in A \to x \in B\)

It implies \(B \subset C\), \(x \in B \to x \in C\)

Now, \(x \in A \to x \in B\) and \(x \in B \to x \in C\) for a particular X.

By transitivity of implication,

It means that \(x \in A \to x \in C\).

Hence, A is a subset of C.

B is a subset of C A is a subset to B

A is a subset to C

Thus, it is shown that A is a subset of C or \(A \subseteq C\).