Hi everyone,

The review sheet for Exam #2, taking place on Tuesday, 10/25, has been posted under `Classroom Resources/Exam Reviews`

. It is unchanged from the previous year. Let me know if you have any questions.

Best,

Prof. Reitz

Skip to the content

Hi everyone,

The review sheet for Exam #2, taking place on Tuesday, 10/25, has been posted under `Classroom Resources/Exam Reviews`

. It is unchanged from the previous year. Let me know if you have any questions.

Best,

Prof. Reitz

- Proof by structural induction November 28, 2021Let BExpr be the variety of all boolean expressions that are defined by the following grammar: I have two recursive functions numbinexprs and numexprs. numbinexprs returns the number of binary subformulas when given a boolean expression, and numexprs returns the number of subformulas. I am supposed to prove the following statement by structural induction: numbinexprs(b) […]yzarc
- How to prove an arbitrary equivalence relation November 28, 2021I got the following definition: $F$ and $G$ are propositional formulas, which are called resemblant, or in symbols $F \asymp G$, if there exists at least one allocation, such that $\beta(F) = \beta(G)$. Now, while the definition is easy, I would like to find out if $\asymp$ is reflexive, transitive and symmetric, but I do […]F.V.
- Count the number of part expressions of an arithmetic expression November 28, 2021The problem: Define the recursive function numexprs that returns the number of part expressions when given an arithmetic expression. Note that even bool is a part expression.(Individual parentheses are not counted as part expressions.) Evaluate your function on the expression below, and verify that it returns the expected value of 8. I don't understand how […]yzarc
- Does specification + reflection prove ZF - extensionality - foundation? November 28, 2021Assume a first-order context. Let $\mathrm{tran}(x)$ denote that $x$ is transitive: $$\mathrm{tran}(x) \leftrightarrow (\forall y \in x) (\forall z \in y) (z \in x)$$ Let $\mathrm{suptran}(x)$ denote that $x$ is supertransitive: $$\mathrm{suptran}(x) \leftrightarrow (\forall y \in x) (\forall z \subseteq y) (z \in x)$$ Consider the following axiom schemas of reflection: \begin{align} & \phi \to […]user76284
- Arc consistency with domain splitting for a graph coloring problem November 28, 2021Consider the graph coloring problem. Below is a graph and we want to color the nodes using one of the three colors Red, Green, or Blue. The constraint is that the two neighboring nodes cannot have the same color. See the graph here (a) Specify the variables, domains of each variable, and the constraint network. […]Taylor
- Does the absolute fragment of second-order logic satisfy a strong Lowenheim-Skolem property? November 28, 2021Let $\mathsf{SOL_{abs}}$ be the "forcing-absolute" fragment of second-order logic - that is, the set of second-order formulas $\varphi$ such that for every (set) forcing $\mathbb{P}$ and every (set-sized) structure $\mathfrak{A}$ we have $$\mathfrak{A}\models\varphi\quad\iff\quad\Vdash_\mathbb{P}(\mathfrak{A}\models\varphi).$$ Note that this is in fact definable since the second-order theory of a structure in $V_\kappa$ is calculated, uniformly, in $V_{\kappa+1}$; by […]Noah Schweber
- If $\vdash A$, then $\vdash A[x:=t]$ November 27, 2021Let $A$ be a formula, $x$ a variable, $t$ a term and $\Gamma$ a set of formulas. If $\Gamma\vdash A$ and $x$ is not a free variable of some open assumption, then $\Gamma\vdash A\to\forall_xA$ and $\vdash \forall_xA\to A[x:=t]$ by the natural deduction rules. Thus, $\Gamma\vdash A[x:=t]$. However, I was just told that $$\tag{1}\text{If }\vdash A\text{, […]Filippo
- What is the best way to transform a specific literal expression in a mathematical expression using logical notation? November 27, 2021I am having a hard time finding the correct solution for the following problem, right now. "New trade agreement means Dollar and Yuan will rise and fall together" How should I express this statement mathematically? Given that: D: the Dollar is strong, Y: the Yuan is strong T:New US-China trade agreement is signed Which one […]Malik Durmus
- A counterexample to $(A\to B)\to\forall_xA\to\forall_xB$ November 27, 2021Let $A$, $B$ be formulas and $x$ a variable. It seems like the formula \begin{equation}\tag{1} (A\to B)\to\forall_xA\to\forall_xB \end{equation} can only be derived (using the natural deduction rules) if $x$ is not a free variable of $A\to B$. However, based on my intuition, I expected that $(1)$ is derivable for all formulas and variables. Maybe a […]Filippo
- Proving Tautological Equivalence with an Inductive Set November 27, 2021Let C = β©{A | A is ({A1}, {ΞΎβ})-inductive}. Show that any element of C is a tautology or tautologically equivalent to A1. By definition of an inductive set, I know that A1 must be a subset of C by virtue of the ΞΎβ function. And I know that to prove any element of C […]lemurs63

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