Explore Discrete Math Examples

Example: Partitions of Sets
a. Let ${\displaystyle A=\{1,2,3,4,5,6\},A_1=\{1,2\},A_2=\{3,4\}}$ and ${\displaystyle A_3=\{5,6\}}$. Is ${\displaystyle \{A_1,A_2,A_3\}}$ a partition of A?
b. Let Z be the set of all integers and let:
${\displaystyle T_0=\{n\in Z\mid n=3k,\text{for some integer}k\}}$
${\displaystyle T_1=\{n\in Z\mid n=3k+1,\text{for some integer}k\}}$, and
${\displaystyle T_2=\{n\in Z\mid n=3k+2,\text{for some integer}k\}}$
Is ${\displaystyle \{T_0,T_1,T_2\}}$ a partition of Z?

Discrete Math Question
Negate the following(in English) :
a) If it is snowing, then it is cold.
b) If ${\displaystyle x\in (0,1)}$, then ${\displaystyle x^2<x}$.
c) If ${\displaystyle x\in T}$, then ${\displaystyle x\in S}$ and ${\displaystyle x\in U}$. (Here S,T,U are sets).

Prove that discrete math:

1) If $a|b\text{and}a|c\text{then}a|(b^2-4c)$
2) If $a|b\text{then}{a}^2|b^2$

Which one of the following quantifiers and predicates expresses the following statement, "There is a student in this lecture who taken at least one course in Discrete Math."?
a) ${\displaystyle \mathrm{\exists }x\mathrm{\exists }yP(x,y)}$ where ${\displaystyle P(x,y)}$ is "x has taken y," the domain for x consists of all student in this class, and the domain for y consists of all Discrete Math lectures.
b) ${\displaystyle \mathrm{\exists }x\mathrm{\exists }yP(x,y)}$ where ${\displaystyle P(x,y)}$ is "x has taken y," the domain for x consists of all Discrete Maths lectures, and the domain for y consists of all student in this class.

c) ${\displaystyle \mathrm{\forall }x\mathrm{\forall }yP(x,y)}$, where ${\displaystyle P(x,y)}$ is "x has taken y," the domain for x consists of all student in this class, and the domain for y consists of all Discrete Math lectures.
d) ${\displaystyle \mathrm{\exists }x\mathrm{\forall }yP(x,y)}$, where ${\displaystyle P(x,y)}$ is "x has taken y," the domain for x consists of all student in this class, and the domain for y consists of all Discrete Math lectures.