EF Converting language to symbols

Section 1.2 Converting language to symbols

As we have already begun to do, we will use letters to represent (possibly variable) logical statements and substatements. To complete the conversion from verbose language to compact symbolism, we will introduce symbols to represent the Five basic connectives.

negation of $A$: 🔗
$\neg A$
conjuction of $A$ and $B$: 🔗
$A \land B$
disjunction of $A$ and $B$: 🔗
$A \lor B$
conditional where $A$ implies $B$: 🔗
$A \to B$
biconditional involving $A$ and $B$: 🔗
$A \leftrightarrow B$

🔗

Using variables to represent statements and the above symbols to represent connectives allows us to isolate the task of analyzing logical structure, without being distracted or influenced by the content of the statements.

🔗

Warning 1.2.1.

🔗

In mathematics, the symbol

\to

is also used in function notation; you will need to determine from the context which role this symbol is playing.

🔗

Example 1.2.2. Translating English language into symbolic language.

🔗

Consider the statement “if we are outside and we get wet then it is raining.” Assign statement variables:

\begin{aligned} A & = “we are outside,” & B & = “we get wet,” & C & = “it is raining.” \end{aligned}

🔗

Then symbolically, the statement can be written

A \land B \to C .

🔗

Remark 1.2.3.

🔗

Using substatement variables is not the same as using free variables. You should think of substatement variables as placeholders for specific logical statements which, by definition, can each be determined to be either true or false.

Elementary Foundations: An introduction to topics in discrete mathematics

Search Results:

Section 1.2 Converting language to symbols

Warning 1.2.1.

Example 1.2.2. Translating English language into symbolic language.

Remark 1.2.3.