The student's question involves logical propositions which are a part of mathematics, specifically dealing with statements about two subjects' states or actions and combining them with logical operators.
The student's question is related to logic, a branch of mathematics. They are describing different logical statements involving two subjects, Eche and Ethan, and their actions or states: being happy and dancing. Let's break down the statements into logical components:
Eche is happy and Ethan is not dancing.
Eche is happy and Ethan is dancing.
Eche is happy if and only if Ethan is dancing.
The first two statements are simple conjunctions, combining two conditions with an 'and' operator. The last statement is a biconditional, indicating that both conditions must either both be true or both be false. This is a classic example used to teach logical propositions and truth tables.