Understanding these basic definition and statement~

• If $x$ belongs to a class $A$, we write $x \in A$ and read as ” $x$ is an element of $A . “$ Otherwise, we write $x \notin A$.
• If $A$ and $B$ are sets, then $A \subseteq B(” A$ is a subset of $B$ ” or “A is contained in $B “$ ) means that each element of $A$ is also an element of $B$. Sometimes we write $B \supseteq A$ (“B contains A”) instead of $A \subseteq B$.
• We say two sets $A$ and $B$ are equal, written $A=B$, if $A \subseteq B$ and $B \subseteq A$.
• Any statement $S$ has a negation $\sim S($ “not $S$ “) defined by
  $\sim S$ is true if $S$ is false and $\sim S$ is false if $S$ is true.
• Let $P(x)$ denote a property $P$ of the object $x$. We write $\exists$ for the quantifier “there exists.” The expression
  $$
  \exists x \in X: P(x)
  $$
  means that “there exists (at least) one object $x$ in the class $X$ which has the property $P . “$ The symbol $\exists$ is called the existential quantifier.

• We use the symbol $\forall$ for the quantifier “for all.” The expression
  $$
  \forall x \in X: P(x)
  $$
  has the meaning “for each object $x$ in the class $X, x$ has property $P . “$ The symbol $\forall$ is called the universal quantifier (or sometimes the general quantifier).
• We use the symbol := to mean “is defined by.” We take $x:=y$ to mean that the object or symbol $x$ is defined by the expression $y$.
• Note that for negation of a statement we have:
  (i) $\sim \sim A:=\sim(\sim A)=A$
  (ii) $\sim(A$ and $B)=(\sim A)$ or $(\sim B)$
  (iii) $\sim(A$ or $B)=(\sim A)$ and $(\sim B)$
  (iv) $\sim(\forall x \in X: P(x))=(\exists x \in X: \sim P(x))$
  (v) $\sim(\exists x \in X: P(x))=(\forall x \in X: \sim P(x))$.
• Let $A$ and $B$ be statements. $A$ implies $B$ will be denoted by $A \Rightarrow B$. If $A$ implies $B$, we take this to mean that if we wish to prove $B$, it suffices to prove $A(A$ is a sufficient condition for
  B).
• The equivalence $A \Leftrightarrow B$ (“A and $B$ are equivalent” or ” $A$ if and only if $B$,” often written $A$ iff $B$ ) of the statements $A$ and $B$ is defined by
  $$
  (A \Leftrightarrow B):=(A \Rightarrow B) \text { and }(B \Rightarrow A)
  $$
  $A$ is a necessary and sufficient condition for $B$, or vice versa.

• The statement $\sim B \Rightarrow \sim A$ is called the contrapositive of the statement $A \Rightarrow B$. In standard logic practices, any statement is considered equivalent to its contrapositive. It is often easier to prove a statement’s contrapositive instead of directly proving the statement itself.
• To prove $A \Rightarrow B$ by contradiction, one supposes $B$ is false (that $\sim B$ is true). Then, also assuming that $A$ is true, one reaches a conclusion $C$ which is already known to be false. This contradiction shows that if $A$ is true $\sim B$ cannot be true, and hence $B$ is true if $A$ is true.
• Given two sets $A$ and $B$, we define $A \cup B$ (“the union of $A$ with $B$ “) as the set
  $$
  A \cup B:={x: x \in A \text { or } x \in B \text { or both }}
  $$
  When speaking about unions, if we say $x \in A$ or $x \in B$ it also includes the possibility that $x$ is in both $A$ and $B$.
• We define $A \cap B$ (“the intersection of $A$ with $B$ “) as the set
  $$
  A \cap B:={x: x \in A \text { and } x \in B} \text {. }
  $$

• Let $A$ and $B$ be subsets of $X .$ Then
  $$
  A \backslash B:={x \in X: x \in A \text { and } x \notin B}
  $$
  is the relative complement of $B$ in $A$. When the set $X$ is clear from the context we write also
  $$
  A^{c}:=X \backslash A
  $$
  and call $A^{c}$ the complement of $A$.
• If $X$ is a set, then so is its power set $\mathcal{P}(X)$. The elements of $\mathcal{P}(X)$ are the subsets of $X$. Sometimes the power set is written $2^{X}$ for a reason which is made clear in Problem $2.8$.
• Let $f: X \rightarrow Y$ be a function, then
  $$
  i m(f):={y \in Y ; \exists x \in X: y=f(x)}
  $$
  is called the image of $f$. We say $f$ is surjective (or onto) if $i m(f)=Y$, injective (or one-to-one) if $f(x)=f(y)$ implies $x=y$ for all $x, y \in X$, and $f$ is bijective if $f$ is both injective and surjective.
• If $X$ and $Y$ are sets, the Cartesian product $X \times Y$ of $X$ and $Y$ is the set of all ordered pairs $(x, y)$ with $x \in X$ and $y \in Y$.
• Let $X$ be a set and $\mathcal{A}=\left{A_{i}: i \in I\right}$ be a family of sets and $I$ is an index set. Intersection and union of this family are given by
  $$
  \bigcap_{i \in I} A_{i}=\left{x \in X ; \forall i \in I: x \in A_{i}\right}
  $$
  and
  $$
  \bigcup_{i \in I} A_{i}=\left{x \in X ; \exists i \in I: x \in A_{i}\right}
  $$

• Inverse image of $B$ under $f$ (or pre-image of $B$ ), $f^{-1}(B)$ defined as
  $$
  f^{-1}(B)={x \in X: f(x) \in B} .
  $$
  Note that we can form $f^{-1}(B)$ for a set $B \subset Y$ even though $f$ might not be one-to-one or onto.
• We will use standard notation, $\mathbb{N}$ for the set natural numbers, $\mathbb{Z}$ for the set of integers, $\mathbb{Q}$ for the set rational numbers, and $\mathbb{R}$ for the set real numbers. We have the natural containments:
  $$
  \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}
  $$
• Two sets $A$ and $B$ have the same cardinality if there is a bijection from $A$ to $B$. In this case we write $A \sim B$. We say $A$ is countable if $\mathbb{N} \sim A$. An infinite set that is not countable is called an uncountable set.

Try the following problem to test yourself！

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