Logic is the foundation to formulate proofs and to understand the language of mathematics.

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Concept |
Content |

Sets |
Sets are the basic building blocks for a lot of mathematics. In order to rigorously define numbers and doing real analysis, we need to know how to work with sets. |

Logical Deduction |
How to get new true proposition from other true propositions. |

Convergence |
Convergent sequences have a well-defined limit. |

Monotonicity of Limits
and Sandwich Theorem |
Taking limits respects weak inequalities and sandwiching a sequence by two converging sequences reveals its limit. The technique of sandwiching can be used to determine limits via nested intervals. |

Heine-Borel
Theorem |
The theorem connecting the concept of compactness with boundedness and closedness. |

Injectivity, Surjectivity,
Bijectivity |
These are important notions for maps. |

Bolzano-Weierstrass
Theorem |
Every bounded sequence has at least one converging subsequence. |

Basic logic is something, we usually accomplish intuitively right.
However, in mathematics we have to define it in an unambiguous way and
it may differ a little bit from the everyday logic. It is very important
and useful to bring into our attention some of the basic rules and
notations of logic. For Computer Science students, logic is considered
in more detail in other courses.

Let us start with a definition:

**Definition 1** (logical statement,
proposition). *A **logical statement* (or *proposition*)
is a statement, which means a meaningful declarative sentence, that is
either true or false.

Instead of *true*, one often writes \(T\) or \(1\) and instead of *false*, one
often writes \(F\) or \(0\).

Not every meaningful declarative fulfils this requirement. There are
opinions, alternative facts, self-contradictory statements, undecidable
statements and so on. In fact, a lot of examples here, outside the
mathematical world, work only if we give the words unambiguous
definitions which we will implicitly do.

**Example 2**. Which of these are logical
statements?

Hamburg is a city.

\(1 + 1 = 2\).

The number \(5\) is smaller than
the number \(2\).

Good morning!

\(x + 1 = 1\).

Today is Tuesday.

The last two examples are not logical statements but so-called
predicates and will be considered later. For given logical statements,
one can form new logical statements with so-called *logical
operations*. In the following, we will consider two logical
statements \(A\) and \(B\).

**Definition 3** (Negation \(\neg A\) (“not \(A\)”)). *\(\neg A\) is true if and only if \(A\) is false.*

\[\mbox{ Truth table }\qquad
\begin{array}{c|c}
A & \neg A\\ \hline
T & F\\
F & T
\end{array}\]

Sometimes you will see the notation \(\sim\! A\) instead of \(\neg A\).

**Example 4**. What are the negations of the following
logical statements?

The wine bottle is full.

The number \(5\) is smaller than
the number \(2\).

All students are in the lecture hall.

**Definition 5** (Conjunction \(A \wedge B\) (“\(A\) and \(B\)”)). *\(A
\wedge B\) is true if and only if both \(A\) and \(B\) are true.*

\[\mbox{ Truth table }\qquad
\begin{array}{cc|c}
A & B & A \wedge B\\ \hline
T & T& T\\
T & F & F\\
F & T & F\\
F & F & F
\end{array}\]

**Definition 6** (Disjunction \(A \vee B\) (“\(A\) or \(B\)”)). *\(A
\vee B\) is true if and only if at least one of \(A\) or \(B\) is true.*

\[\mbox{ Truth table }\qquad
\begin{array}{cc|c}
A & B & A \vee B \\ \hline
T & T& T\\
T & F & T\\
F & T & T\\
F & F & F
\end{array}\]

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