Currently I am reading Introduction to Set Theory by Karel Hrbacek.
Background: First the author stated the axiom of pairing and the axiom of union as follows:
Axiom of union: $\forall S:\exists U: \forall x: \big(x\in U\iff \exists A\in S: x\in A\big)$
Axiom of pairing: $\forall A: \forall B: \exists C: \forall x:\big(x\in C\iff x=A\vee x=B\big)$
Then by proving the uniquess of the sets $U$ and $C$, respectively, we can introduce The union of two sets as a set theoretic operation.
My problem: The author claim is " The axiom of union is much more powerful; it enables us to form unions of not just two, but of any, possibly infinite, collection of sets."
My doubt: Should it not be/Doesn't he mean The axiom of pairing instead?
Short answer: No, the author is correct.
The Axiom of Pairing helps you to define sets $\{A,B\}$ whenever $A,B$ are sets. Then, by the Axiom of Union, you can form the set $\bigcup\{A,B\}$, usually denoted $A\cup B$.
You can now continue this process to form any union of finitely many sets, say $A_1\cup\cdots\cup A_n$. However, using this strategy, you will not be able to make the jump to the union of infinitely many sets.
In fact, in order to form the union $\bigcup_{i\in I} A_i$, where $I$ is some indexing set, and all $A_i$ are just some arbitrary sets, you proceed as follows: Use the Axiom Schema of Replacement (see later in the book) to form the collection $\mathcal C=\{A_i\mid i\in I\}$, then form its union $\bigcup\mathcal C$ using the Axiom of Union, which is precisely what we mean by $\bigcup_{i\in I} A_i$. As $I$ could be infinite, the union could be as well.