Logic
Answer to the last homework question: Since the funny face is drawn by only one girl, there can be three possibilities: 1) D did it; 2) B did it; and 3) neither D nor B, but some other girl did it. Given that M is a knave, what he said (''it is B instead of D who did it'') must be false, so 2) cannot be the case, but 1) or 3) may happen. We can then conclude that B definitely did not do it, but we are not sure about D: she did it in case 1), and did not do it in case 3).Next Question: K really wants to find out if D is the girl who drew the funny face. So he brings the case to the court run by the girls. Here D is the D(efendant), R and B are witnesses, and M is the judge. A trial took place and we have obtained the following court record:
- R: D once told us she drew the funny face on K's back.
- D: R is lying, I never said that.
- B: D once said she did not draw the funny face.
- M asks D: Is B telling the truth?
- D said either yes or no, but this part of the record is blurred, so we cannot tell which answer she gave to the judge.
- M: Now I know whether D is guilty or innocent of drawing that funny face.
D, R, B can be a knight who always tells the truth or a knave who always lies.
Question: 1) what did D tell the judge? 2) is D innocent or guilty?
Like the analysis of the last week's problem, here we can answer these questions by considering the two possible answers (yes or no) separately and fill out the truth table in each case.
First, B and D cannot both tell the truth. For otherwise, D would have said "I drew the funny face" and "I didn't draw the funny face". One of them must be true and the other must be false, which is not possible since D is either a knight who never lies or a knave who never tells truth.
Thus there can be three situations: 1) D only said "I drew the funny face"; 2) D only said " I didn't drew the funny face", and 3) D said nothing.
Thus there can be three situations: 1) D only said "I drew the funny face"; 2) D only said " I didn't drew the funny face", and 3) D said nothing.
If D's answer to judge's question is no, i.e., B did not tell the truth and D never said "I didn't do it". Then we have the following truth table:
| actual case | D said "I did it" | D said "I didn't do it" | ||
|---|---|---|---|---|
| R: D said that | D: I didn't | B: D said that | D: I didn't | |
| D only said "I did it" | T | F | F | T |
| D only said "I didn't do it" | F | T | T | F |
| D said nothing | F | T | F | T |
Since D is either a knight or a knave, the first two cases, in which D told a truth and a lie, are not possible. So both R and B are knaves, and D is a knight and she said nothing before. In this case, the judge has no way to know whether D actually did it.
Homework: Suppose that the answer from D to the judge M is yes, i.e., D agree with B that she said "I didn't do it'' before. Use a similar truth table to figure out if D is guilty or innocent.
Coordinate Plane
Introduction to two axesWe have been working on the number line. Today we started looking at the coordinate plane
We still have an origin, but now we have X and Y axes.
We name a point with (x, y). These are called the 'cartesian coordinates'.
Each student should be able to draw these on graph paper where one square = one unit
There are four quadrants
Which quadrant are the following points in? A(1, -3) B(-3, 1), ...
Draw the points (1, 1), (1, -1), (-1, 1), (-1, -1).
What shape does it make? (most thought it made a square ... but it made an "N")
Homework
1. Plot the function y = |x|Hint: first fill out the table
| x | y = |x| |
|---|---|
| -4 | 4 |
| -3 | |
| -2 | |
| -1 | |
| 0 | |
| 1 | |
| 2 |
2. Plot and label the points
B(-3, 4)
C(-5, 4)
D(-6, 2)
E(-5, -1)
F(-3, -3)
G(0, 6)
H(3, -3)
I(5, -1)
J(6,2)
K(5,4)
L(3,4)
M(2,3)
N(0,1)
