Logical Statements and Operations
Logic is the foundation to formulate proofs and to understand the language of mathematics.
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Study Logical Statements and Operations
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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