Maths 120 Semester 2, 2020 Assignment 1

1.（命题逻辑和真值表。）在课堂上，我们研究了逻辑运算^，_，），，

q
T
F
T
F F F

T
F
T
F
（a）证明复合命题p _ q和p（（（（p？q）（q）

（b）在（a）中，您证明了我们可以创建与逻辑连接词_等效的东西

1个
2.（纠正错误。）您的朋友正在准备学期中的考试！为了练习，他们尝试了

n，n + 50和n + 100是3的倍数。

n是整数，则恰好是三个值n，n + 50，n + 100中的一个是整数

3的倍数，则n必须是整数。

n = 7不是3的倍数，并且n + 100 = 107 = 35·3 + 2也是

（a）找到您朋友证明中的所有逻辑错误1，并解释为什么错误

（b）为您的朋友正在解决的问题写一个有效的解决方案。
3.（证明方法。）假设您有一个3⇥3的网格，并且该网格中的每个正方形都有一个

b
d
c
f

4 of its neighbors.
Similarly, the value a is equal to b+d
2 (and
so on/so forth for all of the other values in
our grid.)
What are the possible values in all of the other squares in our grid?
4. (Set operations.)
(a) Let A, B be a pair of subsets of Z. Prove that the following equation always holds:
A \ B = Z \

Z \ A

[
Z \ B

(b) In a sense, (a) shows us that you can “create” the intersection operation \ with just [
and \. That is: if you know how to perform the [ operation and the \ operation, you
can make A \ B without ever actually having to calculate an intersection!
Can you do the same with \? That is: if you have two sets A, B of real numbers, and
can combine them along with Z using \ and [, can you create A \ B? Either find a way
to do this, or explain why this is impossible.
1A logical mistake is an error in the reasoning, not the conclusion. That is: if you want to explain why someone’s
argument is false, you can’t just say “your argument is bad because its conclusion is false;” the person you’re talking
to will just say that the conclusion is actually true and that they’ve proven it! Instead, you want to find a flaw in
their reasoning; i.e. a place where they’ve switched A =) B for B =) A or forgotten an important case in their
proof, or something like that.