# Predicate logic deciding whether atomic formulae hold in interpretations

Consider the formula $$\varphi$$ of First-order logic defined as

$$\forall x\forall y((B(x,y) \land B(y,x)) \rightarrow (A(x)\land C(y)))$$

State whether it holds in the following interpretations:

1. The domain of discourse is the set of natural numbers $$\mathbb{N}$$ with $$B=\{(a,b) \in \mathbb{N} \times \mathbb{N}: a \le b\}$$ and $$A=C=\mathbb{N}$$
2. The domain of discourse is a class of 30 schoolchildren, 2 of whom are twins, with $$B$$ being the set of pairs of distinct schoolchildren of the same age (in whole months), $$A$$ being the complete set of 30 schoolchildren, and $$C$$ being a subset of 29 schoolchildren with one of the twins removed.
3. The domain of discourse is the set of rational numbers $$\mathbb{Q}$$ with $$B = \{(a,b) \in\mathbb{Q} \times \mathbb{Q}: a < b \}, \ A = \emptyset,$$ and $$C = \{0\}$$

My main confusion comes from understanding how the sub-formulae work and how should they be interpreted in the situation. For example, what exactly does $$A(x)$$ mean in $$(ii)$$, likewise with $$C(y)$$. What is the best way to understand and approach questions like these?

• It appears some deficient writer is using A(x) to mean x in A. – William Elliot May 13 at 21:38
• @WilliamElliot Huh? That's not "deficient" at all - we identify $n$-ary predicates with subsets of the $n$th Cartesian power as standard. – Noah Schweber May 13 at 21:50
• Collectively standard deficiency. – William Elliot May 14 at 3:31