Number Systems

$\mathbb N=$ the set of positive integers (not including zero)

$\mathbb Z=$ the set of integers

$\mathbb Q=$ the set of rational numbers

$\mathbb R=$ the set of real numbers

$\mathbb C=$ the set of complex numbers

Logic

Iff: if and only if

If A and B are mathematical assertions, and -A and -B are their negations, the statement “A implies B” is logically equivalent to the contrapositive statement (逆否命题, declaración contrapositive) “-B implies -A.”

Reductio ad absurdum (归谬法): assume both A and -B and derive a contradiction.

$=$: logical identity.

Sets

$\in$: an element/object belongs to a set.

$\notin$: an element/object does not belong to a set.

$\subseteq$: subset.

$\subset$, $\subsetneq$, or $latex\subsetneqq$: proper subset.

$\emptyset$: empty set.

$\mathcal P(A)$: the power set of $A$, i.e. the set of all subsets of $A$.

$\mathcal P(X)=\{E: E\subset X\}$: the family (collection) of all subsets of a set $X$.

$\cup_{E\in\mathcal E} E=\{x: x\in E$ for some $E\in\mathcal E\}$: the union.

$\cap_{E\in\mathcal E} E=\{x: x\in E$ for all $E\in\mathcal E\}$: the intersection.

$\mathcal E=\{E_\alpha: \alpha\in A\}=\{E_\alpha\}_{\alpha\in A}$: the indexed family of sets.

$\cup_{\alpha\in A} E_\alpha$: the union of indexed sets.

$\cap_{\alpha\in A}E_\alpha$: the intersection of index sets.

If $E_\alpha\cap E_\beta=\emptyset$ whenever $\alpha\not=\beta$, the sets $E_\alpha$ are defined as disjoint. Note “disjoint collection of sets”=”collection of disjoint sets”, and “disjoint union of sets”=”union of disjoint sets”.

$\{E_n\}_{n=1}^\infty$ or $\{E_n\}_1^\infty$: families of sets indexed by $\mathbb N$.

$\lim \sup E_n=\cap_{k=1}^\infty\cup_{n=k}^\infty E_n=\{x: x\in E_n$ for infinitely many $n \}$: the limit superior.

$\lim \inf E_n=\cup_{k=1}^\infty\cap_{n=k}^\infty E_n=\{x: x\in E_n$ for all but finitely $n \}$: the limit inferior .

$E\setminus F=\{x: x\in E\}$ and $x\not=F$: the difference of sets $E$ and $F$.

$E\triangle F=(E\setminus F)\cup(F\setminus E)$: the symmetric difference of sets $E$ and $F$.

$E^c=X\setminus E$: the complement.

$(\cup_{\alpha\in A}E_\alpha)^c=\cap_{\alpha\in A}E^c_\alpha$$(\cap_{\alpha\in A}E_\alpha)^c=\cup_{\alpha\in A}E^c_\alpha$: deMorgan’s laws.

$X\times Y$: Cartesian product, i.e. the set of all ordered pairs $(x, y)$ such that $x\in X$ and $y\in Y$.

$R$: a relation from $X$ to $Y$, which is a subset of of $X\times Y$, so $xRy\Rightarrow (x, y)\in R$.

There are three important types of relations: the equivalence relations, the orderings, and the mappings.

“Forgiveness is the most powerful thing that you can do for your physiology and your spirituality. Yet, it remains one of the least attractive things to us, largely because our egos rule so unequivocally. To forgive is somehow associated with saying that it is all right, that we accept the evil deed. But this is not forgiveness. Forgiveness means that you fill yourself with love and you radiate that love outward and refuse to hang onto the venom or hatred that was engendered by the behaviors that caused the wounds.” ——Wayne Dyer

To be continued tomorrow.

References: [1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 1-3.

[2] James R. Munkres, Topology: A First Course, page 3-13.

[2] Purpose Fairy’s 21-Day Happiness Challenge, https://s3.amazonaws.com/Gift1/Free+eBook+-+PurposeFairy’s+21-Day+Happiness+Challenge.pdf