1.1.3.1.
Answer.
These answers are not unique.
- \(\displaystyle 8, 15, 22, 29\)
- \(\displaystyle \textrm{apple, pear, peach, plum}\)
- \(\displaystyle 1/2, 1/3, 1/4, 1/5\)
- \(\displaystyle -8, -6, -4, -2\)
- \(\displaystyle 6, 10, 15, 21\)
\(p\) | \(q\) | \(\neg (p\land q ) \) |
0 | 0 | 1 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
\(p\) | \(q\) | \(p \land (\neg q)\) |
0 | 0 | 0 |
0 | 1 | 0 |
1 | 0 | 1 |
1 | 1 | 0 |
\(p\) | \(q\) | \(r\) | \(p\land q\) | \((p \land q)\land r\) |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 0 |
1 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 0 | 0 |
1 | 1 | 0 | 1 | 0 |
1 | 1 | 1 | 1 | 1 |
\(p\) | \(q\) | \(r\) | \(d\) |
0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 1 | 0 | 0 |
0 | 1 | 1 | 1 |
1 | 0 | 0 | 0 |
1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 |
\(p\) | \(q\) | \(\text{ }\neg p\lor \neg q\) |
0 | 0 | 1 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
\(p\) | \(q\) | \(r\) | \(s\) | \(f\) |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 1 |
0 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 1 |
0 | 1 | 0 | 1 | 1 |
0 | 1 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 1 |
1 | 0 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 | 1 |
\(p\) | \(q\) | \(r\) | \(x\) |
0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 1 | 0 | 0 |
0 | 1 | 1 | 0 |
1 | 0 | 0 | 1 |
1 | 0 | 1 | 1 |
1 | 1 | 0 | 0 |
1 | 1 | 1 | 1 |
\(p\) | \(q\) | \(r\) | \(x\) |
0 | 0 | 0 | 1 |
0 | 0 | 1 | 1 |
0 | 1 | 0 | 0 |
0 | 1 | 1 | 0 |
1 | 0 | 0 | 1 |
1 | 0 | 1 | 1 |
1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 |
(1) | \((x,y)\in A \times (B\cap C)\) | Premise |
(2) | \(x \in A\) | (1), definition of Cartesian Product |
(3) | \(y \in B \cap C\) | (1), definition of Cartesian Product |
(4) | \(y \in B\) | (3),definition of intersection |
(5) | \((x,y) \in A\times B\) | (2),(4), definition of Cartesian Product |
(6) | \(y \in C\) | (3),definition of intersection |
(7) | \((x,y) \in A\times C\) | (2),(6), definition of Cartesian Product |
(8) | \((x,y) \in (A\times B)\cap (A\times C) \) | (5),(7), definition of Cartesian Product \(\quad \blacksquare\) |
(1) | \(x_0 \in A \cap C\) | Negated Conclusion - \(A\cap C\) non-empty |
(2) | \(x_0 \in A\) | (1), definition of intersection |
(3) | \(A \subseteq B\) | Premise |
(4) | \(x_0 \in B\) | (2),(3),definition of subset |
(5) | \(x_0 \in C\) | (1), definition of intersection |
(6) | \(x_0 \in B \cap C\) | (4),(5), definition of intersection |
(7) | \(B \cap C= \emptyset\) | Premise |
(7) | Contradiction | (6),(7) \(\quad \blacksquare\) |
Part | reflexive? | symmetric? | antisymmetric? | transitive? |
i | yes | no | no | yes |
ii | yes | no | yes | yes |
iii | no | no | no | no |
iv | no | yes | yes | yes |
v | yes | yes | no | yes |
vi | yes | no | yes | yes |
vii | no | no | no | no |
R : \(x r y\) if and only if \(\lvert x -y \rvert = 1\) |
S : \(x s y\) if and only if \(x\) is less than \(y\text{.}\) |
Step | \(S=T\text{?}\) | \(S\) | \(T\) |
---|---|---|---|
1,2 | - | \(\left( \begin{array}{ccccc} 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ \end{array} \right)\) | \(\left( \begin{array}{ccccc} 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ \end{array} \right)\) |
3a | No | \(\left( \begin{array}{ccccc} 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ \end{array} \right)\) | \(\left( \begin{array}{ccccc} 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ \end{array} \right)\) |
3b | No | \(\left( \begin{array}{ccccc} 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ \end{array} \right)\) | \(\left( \begin{array}{ccccc} 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ \end{array} \right)\) |
3c | Yes | \(-\) | \(-\) |
Step | Flow-augmenting path | Flow added |
1 | \(\text{Source},A,\text{Sink}\) | 2 |
2 | \(\text{Source}, C,B, \text{Sink}\) | 3 |
3 | \(\text{Source},E,D, \text{Sink}\) | 4 |
4 | \(\text{Source},A,B,\text{Sink}\) | 1 |
5 | \(\text{Source},C,D, \text{Sink}\) | 2 |
6 | \(\text{Source},A,B,C,D, \text{Sink}\) | 2 |