### Logic: Propositional logic

### Negation, conjunction, and disjunction

Now that we know what statements qualify as a proposition in a mathematical sense, we will define operations to construct new propositions from existing ones. These operations are so-called logical operations, and they are indicated by logical operators. We will look at three types of logical operators. First we consider the operator that reverses the assignment of true or false to a proposition.

Negation

The **negation** of a proposition is the opposite of the original proposition. The negation is true exactly when the original proposition is false, and false exactly when the original proposition is true.

Negation is denoted by the *not*-operator: #\neg#. So, if #\blue p# stands for a proposition then its negation is denoted by #\neg{\blue p}#.

**Example**

The negation of the proposition #\blue {\textrm{"It is raining"} }# is given by #\blue{ \textrm{"It is} \textit{ not } \textrm{raining"} }#.

If #\blue {\textrm{"It is raining"}}# is true, then #\blue {\textrm{"It is }\textit{not} \textrm{ raining"}}# is false.

If #\blue {\textrm{"It is raining"}}# is false, then #\blue {\textrm{"It is }\textit{not} \textrm{ raining"}}# is true.

Operator notation: #\neg\blue {\textrm{"It is raining"}}#

Next we consider the composition of two propositions corresponding to the usual conjunctive "and".

Conjunction

The **conjunction** of two propositions is a proposition that is true exactly when both original propositions are true. If at least one of these propositions is false, the conjunction is also false.

Conjunction is denoted by the *and*-operator: #\land#. So, if #\blue {p}# and #\blue q# stand for propositions, then their conjunction is denoted #{\blue p}\land {\blue q}#.

**Example**

The conjunction #\blue {\textrm{"I like beans }\textit{and} \textrm{ carrots"}}# is precisely true when the propositions #\blue {\textrm{"I like beans"}}# and #\blue {\textrm{"I like carrots"}}# are both true.

If at least one of them is false, then #\blue {\textrm{"I like beans }\textit{and} \textrm{ carrots"}}# is false.

Operator notation: #\blue {\textrm{"I like beans"}} \land \blue{\text{"I like carrots"}}#.

As could be expected, next we consider the composition of two propositions which corresponds to the usual disjunctive "or".

Disjunction

The **disjunction** of two propositions is a proposition that is true when at least one of the original propositions is true. The disjunction is only false when exactly both original propositions are false.

Disjunction is denoted by the *or*-operator: #\lor#. So, if #\blue {p}# and #\blue q# stand for propositions, then their disjunction is denoted #{\blue p}\lor {\blue q}#.

**Example**

The disjunction #\blue{\textrm{"I will travel by car }\textit{or} \textrm{ by train"}}# is true when at least one of the propositions #\blue{\textrm{"I will travel by car"}}# or #\blue{\textrm{"I will travel by train"}}# is true. If both of the propositions are false, the proposition #\blue{\text{"I will travel by car }\textit{or} \textrm{ by train"}}# is false.

Operator notation: #\blue{\text{"I will travel by car"}} \lor \blue{\text{"I will travel by train"}}#.

\[\text{I will travel by train or by bike}\]

The proposition "I will travel by train or by bike" contains a disjunction. The two parts are "I will travel by train" and "I will travel by bike", and these two parts are connected by the

*or*-operator.

There is no negation and no conjunction here.

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