Final answer:
The child's ability to associate the word "cat" with the corresponding picture and vice versa demonstrates the equivalence relation of symmetry, indicative of their cognitive and language acquisition progression.
Step-by-step explanation:
The ability of a child to say "cat" when shown a picture of a cat and to point to the picture of a cat when given the prompt "cat" demonstrates the equivalence relation known as symmetry. Symmetry is one aspect of stimulus equivalence, which suggests that if a child can go from a stimulus to a response (e.g., picture of a cat to saying "cat") and then also from the response to the stimulus (e.g., saying "cat" and pointing to the picture of a cat), then those two stimuli are equivalent. It shows that the child understands that the vocalization "cat" and the picture of a cat have the same meaning, so the response is interchangeable.
This symmetry is part of a larger concept in language and cognitive development where children understand abstract concepts through associative learning processes. The example given also reflects on the intricacies of language acquisition where children learn to associate words with their corresponding objects and are able to reverse this association as well.