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First, let us show that the two assumptions are necessary. Here is an example showing what goes wrong when $x = y$. Take $t = x$, $u = 1$, $v = 2$. We have $$x\{x := 1\}\{x := 2\} = 1\{x := 2\} = 1,$$ whereas $$x\{x := 2\}\{x := 1\{x := 2\}\} = 2\{x := 1\} = 2.$$ Next, here is an example showing what goes wrong when $x$ is not free in $v$. Take $t = y$, $... 1 F(x,y,t)⟹ person x can fool person y at time t. For the sake of simplicity propagate negation sign outward by applying De Morgan's law. ∀x∃y∃t(¬F(x,y,t))≡¬∃x∀y∀t(F(x,y,t)) [By applying De Morgan's law.] Now converting ¬∃x∀y∀t(F(x,y,t)) to English is simple. ¬∃x∀y∀t(F(x,y,t))⟹ There does not exist a person who can fool everyone all the time. Which means No ... 0 A problem$L$is$\text{NP}$-complete if$L$is in$\text{NP}$, and$L$is$\text{NP}$-hard (that is,$A\leq_p L$for all$A\in \text{NP}$). Consider the following claims. Claim 1: if$L$is$\text{NP}$-complete and$L\in \text{P}$, then$\text{NP} \subseteq \text{P}$(that is, all problems in$\text{NP}$can be solved in deterministic polynomial time). ... 0 The value of$(a \oplus b \oplus c \oplus d)_i$depends on whether$i$is divisible by$3,4,5,7$, and in particular, by$i \bmod 3\cdot 4\cdot 5\cdot 7$. In order to complete the calculation, you need to know, for each$j \in \{0,\ldots,3\cdot 4\cdot 5\cdot 7 - 1\}$, how many indices$i \in \{1,\ldots,10^{11}\}$satisfy$i \equiv j \pmod{3\cdot 4\cdot 5\cdot ...

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I agree with reinierpost's answer and with Yuval Filmus's and lemontree's comments. Here are some additional points. In English "a car" in "Tom has a car" usually means that there is some car that Tom has. "a car" is not a name of a specific car. That is, it is not a constant. You could treat "car" as a name of an ...

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There is no universal answer. It depends on how you choose to model the universe of discourse you've been assigned to model. We'd normally expect there to be multiple cars that appear as distinct individuals in the model, and we expect the predicate has to be the relationship that expresses which individual owns which individual object, i.e. between ...

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