Presentation of free product is disjoint union of presentations

For two groups
Suppose $$G_1$$ and $$G_2$$ are groups given by presentations. Then, a presentation for the external free product $$G_1 * G_2$$ is given as follows: we first make sure that all the letters for generators in the presentation of $$G_1$$ are different from the letters in the presentation of $$G_2$$ (i.e., we make disjoint the sets of generators). Now, we take the generating set for $$G_1 * G_2$$ as the union of both these generating sets and the relation set for $$G_1 * G_2$$ as the union of the relations for $$G_1$$ and $$G_2$$.

For a collection of more than two groups
The same rule applies: we make pairwise disjoint generating sets for all the groups, then take a disjoint union of the generators and of the relations.

Examples
In the examples, we use $$1$$ to stand for the identity element. To avoid confusion, we already make disjoint the presentations for the two groups for which we are taking the free product.

More examples should be added to illustrate free product with more complicated presentations

Related facts

 * Presentation of direct product is disjoint union of presentations plus commutation relations