posted 11 months ago

\text { (b) Show that if }\left\{\alpha, \gamma_{1}\right\} \vDash \chi \text { and }\left\{\beta, \gamma_{2}\right\} \vDash \chi \text { then }\left\{(\alpha \vee \beta), \gamma_{1}, \gamma_{2}\right\} \vDash \chi

(c) Suppose that I is a set of propositions. Show that if = varphi then there is a finite subset of i c i that i= varphi

(d) Let X be a countably infinite set.

(i) Find a set of propositional variables VAR and a bijection

f: 2^{\mathrm{VAR}} \rightarrow\left\{(A, B, R) \mid A, B \subseteq X, R \subseteq X^{2}\right\}

(ii) Find a set of propositions I so that

V(\Gamma)=\{v \mid v(\varphi)=1 \text { for all } \varphi \in \Gamma\}

corresponds to

P=\{(A, B, R) \mid A, B \subseteq X, R \subseteq((A \cap B) \times(A \cup B)) \cup((A \cup B) \times(A \cap B))\}

under the bijection f. You must prove that f(V(T)) = P.

posted 11 months ago

\langle\{a, b\} ;\{(a, a),(b, b)\}\rangle \vDash \varphi

and

\langle\{a, b\} ;\{(a, a),(b, a)\}\rangle \not \forall \varphi .

) Let varphi be the sentence Jr(f(f(x))) # f(x)). For each of the following sentences, explain if it is satisfiable or not. If it is satisfiable,find a structure which satisfies it, and if not, explain why it is unsatisfiable.

\text { (i) } \varphi \wedge(\exists y \forall x f(x)=y)

\text { (ii) } \varphi \wedge(\forall x \exists y f(x)=y)

\text { (iii) } \varphi \wedge(\exists y \forall x f(y)=x)

\text { (iv) } \varphi \wedge(\forall x \exists y f(y)=x)

(c) Consider the structure

\boldsymbol{X}=\left\langle\mathbb{N} ; f^{\boldsymbol{X}}, 0_{\mathbf{N}}\right\rangle

where fX is the function defined by f*(n) = n + 1.

(i) Show that

\boldsymbol{\Gamma}=\operatorname{Th}(\boldsymbol{X}) \cup\{(\boldsymbol{x} \neq \underbrace{f \cdots f}_{n \text { times }} 0\}

is satisfiable.

(ii) Give an example of a structure y and a substitution s so that y ET[s].

posted 11 months ago

\varphi=((\forall x \exists y(x=f y)) \wedge(\forall x \forall y(f x=f y \rightarrow x=y)))

(i) Give a short explanation of what it means for a structure to satisfy the sentence varphi above.

(iii) For each n > 0, let psin be the sentence

(ii) Show that there are types of countable structures which satisfy p.uncountably many different isomorphism

\forall x(\neg(x=\underbrace{f \cdots f}_{n \text { times }} x))

Show that there are only countably many different isomorphism types of countable structures which satisfy both varphi and psin for all n > 0.

posted 11 months ago

(c) Give natural deduction proofs of the following:

(b) Find a set T = {p1,Y2, Y3, P4} consisting of four propositions so that I is inconsistent and any proper subset of I is consistent.Give a natural deduction proof that I is inconsistent, illustrate this proof with a Venn diagram involving the sets V (y;), and also prove that {varphi1, varphi2, varphi3} is consistent.

\text { (i) }\{\alpha \rightarrow(\neg \beta \rightarrow \gamma), \alpha \rightarrow(\neg \beta \rightarrow \neg \gamma)\} \vdash \alpha \rightarrow \beta

\text { (ii) }\{(\alpha \rightarrow \neg \beta) \rightarrow(\gamma \rightarrow \alpha), \neg \beta, \gamma \rightarrow(\alpha \rightarrow \delta)\} \vdash \gamma \rightarrow \delta

posted 11 months ago

(b) Denote by |X| the cardinality of a set X.

(i) Let A, B, C, and D be sets such that |A| = |B| and |C| =|D]. Prove that

|Ac| = |Bd

(ii) Find a pair of sets A and B so that |A| < |B|, but |AN| =|B]. Find an example where B is countably infinite. (Noproof is necessary.)

(iii) Find two different explicit bijections f, g: 5N → 2N. By explicit, I mean recipes for converting elements of 5N into elements of 2N. Explain your answer graphically in terms of the trees for 2N and 5N.

posted 1 years ago

(a) Compute the product using the definition where Ax is the linear combination of the columns of A using the corresponding entries in x as weights. If the product sun defined, explain why. Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice.

Ах-

B. The matrix-vector Ax is not defined because the number of rows in matrix A does not match the number of entries in the vector x.

The matrix-vector Ax is not defined because the number of columns in matrix A does not match the number of entries in the vector x.

posted 1 years ago

O A. The statement that Hx = c is inconsistent for some c is equivalent to the statement that Hx = c has no solution for some c. From this, all of the statements in the Invertible Matrix Theorem are false, including the statement that Hx = 0 has only the trivial solution. Thus, Hx = 0 has a nontrivial solution.

. The statement that Hx =c is inconsistent for some c is equivalent to the statement that Hx c has a solution for every c. From this, all of the statements in the Invertible Matrix Theorem are true, including the statement that the columns of H form a linearly independent set. Thus, Hx = 0 has an infinite number of solutions.

O C. The statement that Hx = c is inconsistent for some c is equivalent to the statement that Hx = c has a solution for every c. From this, all of the statements in the Invertible Matrix Theorem are true, including the statement that Hx =0 has only the trivial solution.

O B. The statement that Hx = c is inconsistent for some c is equivalent to the statement that Hx = c has no solution for some c. From this, all of the statements in the Inyertible Matrix Theorem are false, including the statement that Hx 0 has only the trivial solution. Thus, Hx =0 has no solution.

posted 1 years ago

6 x_{1}+x_{2}-3 x_{3}=10

8 x_{2}+2 x_{3}=0

Write the system as a vector equation where the first equation of the system corresponds to the first row. Select the correct choice below and fill in any answer boxes to complete your choice.

\begin{array}{l} \text { O. }\left[\left[\begin{array}{lll} x_{1} & x_{2} & x_{3} \end{array}\right]=\square\right. \\ \text { B. }\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]=\square \\ \text { 0 c. } x_{1} \square+x_{2} \square+x_{3} \square= \end{array}

posted 1 years ago

posted 1 years ago