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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?

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  • $\begingroup$ It appears some deficient writer is using A(x) to mean x in A. $\endgroup$ – William Elliot May 13 at 21:38
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    $\begingroup$ @WilliamElliot Huh? That's not "deficient" at all - we identify $n$-ary predicates with subsets of the $n$th Cartesian power as standard. $\endgroup$ – Noah Schweber May 13 at 21:50
  • $\begingroup$ Collectively standard deficiency. $\endgroup$ – William Elliot May 14 at 3:31
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  1. A(x) is the predicate "x is a student in the class".
  2. C(y) is the predicate "y = 0".
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