**Section F**In class, we showed that the two-place Boolean function $ F : \{0,1\} \times \{0,1\} \rightarrow \{0,1\} $, defined by $ F : (a, b) \rightarrow (a + b + ab) \mod 2 $, corresponds to the $\vee$ connective from logic.**(F.1) Logical Connective Definition**Let us define the logical connective $\pitchfork$ using the truth table below. Create a two-place Boolean function (similar to the one given above) that corresponds to the $\pitchfork$ connective. (Just write the function definition.)\[\begin{array}{c|c|c}p & q & p \pitchfork q \\\hlineT & T & F \\T & F & T \\F & T & T \\F & F & T \\\end{array}\]**(F.2) Function Evaluation**Evaluate your function given in (F.1) for each element in its domain.---**Section G**Students (including yourself) from 5 area colleges are at a math conference (there are more than 8 students per college). What is the fewest number of students that you would have to meet in order to guarantee that you meet at least 8 from the same college?---**Section H**Let $\mathbb{O}$ be the set of odd integers and consider the function $\Gamma : \mathbb{Z} \rightarrow \mathbb{O}$ defined by $\Gamma(n) = 2n - 3$. Show that $\Gamma$ is onto, by:1. Finding an element, say $x$ of $\mathbb{Z}$, that gets sent to an arbitrary element $y$ in $\mathbb{O}$ (make sure to show that your element does in fact get sent to $y$).2. Proving that $x$ is in fact an element of $\mathbb{Z}$.

Solution for F: F.1 given truth table is P Q P#Q T T F T F T F T T F F T where # is the connective. Here P and Q satisfied for (P, Q)→~(P∧Q) So the function will be F(a, b)=abF.2 Here function is F(a, b)=ab and elements of domain are {(0, 1)×(0, 1)}=0, 0, 0, 1, 1, 0, 1, 1 So F(0, 0)=0.0=0=1⇒F(0, 0)=1 F(0, 1)=0.1=0=1⇒F(0, 1)=1 F(1, 0)=1.0=0=1⇒F(1, 0)=1 F(1, 1)=1.1=1=0⇒F(1, 1)=0

Math

Advanced Math

[F] In class we showed that the two-place Boolean function F: {0, 1} × {0,1} → {0, 1} defined by F : (a, b) → (a + b+ ab)mod 2 , corresponded to the V connective from logic. (F.1) Let us define the logical connective # as shown in the truth table below. Create a two-place Boolean function (defined by a mod formula like that above) that correspondes to the # connective. (Just write the function definition) p# q T T F T T F T T F F T (F.2) Evaluate your function given in (F.1) for each element in its domain.

[F] In class we showed that the two-place Boolean function F: {0, 1} × {0,1} → {0, 1} defined by F : (a, b) → (a + b+ ab)mod 2 , corresponded to the V connective from logic. (F.1) Let us define the logical connective # as shown in the truth table below. Create a two-place Boolean function (defined by a mod formula like that above) that correspondes to the # connective. (Just write the function definition) p# q T T F T T F T T F F T (F.2) Evaluate your function given in (F.1) for each element in its domain.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Section F**

In class, we showed that the two-place Boolean function $ F : \{0,1\} \times \{0,1\} \rightarrow \{0,1\} $, defined by $ F : (a, b) \rightarrow (a + b + ab) \mod 2 $, corresponds to the $\vee$ connective from logic.

**(F.1) Logical Connective Definition**

Let us define the logical connective $\pitchfork$ using the truth table below. Create a two-place Boolean function (similar to the one given above) that corresponds to the $\pitchfork$ connective. (Just write the function definition.)

\[
\begin{array}{c|c|c}
p & q & p \pitchfork q \\
\hline
T & T & F \\
T & F & T \\
F & T & T \\
F & F & T \\
\end{array}
\]

**(F.2) Function Evaluation**

Evaluate your function given in (F.1) for each element in its domain.

---

**Section G**

Students (including yourself) from 5 area colleges are at a math conference (there are more than 8 students per college). What is the fewest number of students that you would have to meet in order to guarantee that you meet at least 8 from the same college?

---

**Section H**

Let $\mathbb{O}$ be the set of odd integers and consider the function $\Gamma : \mathbb{Z} \rightarrow \mathbb{O}$ defined by $\Gamma(n) = 2n - 3$. Show that $\Gamma$ is onto, by:

1. Finding an element, say $x$ of $\mathbb{Z}$, that gets sent to an arbitrary element $y$ in $\mathbb{O}$ (make sure to show that your element does in fact get sent to $y$).
2. Proving that $x$ is in fact an element of $\mathbb{Z}$.

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