The Proof Page

by D. A. Kouba

Section 1.2- Conditionals and Biconditionals; Mathematically Equivalent Statements



In this section we will introduce two more mathematical connectives. This will allow for a wider and more useful range of propositional forms. In particular, we will now include in our propositional forms, sentences of the form ``If $ P $ , then $ Q $", one of the most useful forms in the study of mathematics.

$ \underline { \rm DEFINITION } $ : Let $ P $ and $ Q $ be propositions. The $ \underline { \rm conditional \ sentence } $ $ P \Rightarrow Q $ (read ``$ P $ implies $ Q $." or ``If $ P $, then $ Q $.") is true whenever $ Q $ is true or $ P $ is false.

The following truth table illustrates all possible truth values for the conditional sentence $ P \Rightarrow Q $.


$ P $ $ Q $ $ P \Rightarrow Q $
$ T $ $ T $ $ T $
$ T $ $ F $ $ F $
$ F $ $ T $ $ T $
$ F $ $ F $ $ T $



$ \underline { \rm DEFINITION } $ : Let $ P $ and $ Q $ be propositions. The $ \underline { \rm biconditional \ sentence } $ $ P \Leftrightarrow Q $ (read ``$ P $ if and only if $ Q $.") is true exactly when $ P $ and $ Q $ have the same truth values.

The following truth table illustrates all possible truth values for the biconditional sentence $ P \Leftrightarrow Q $.


$ P $ $ Q $ $ P \Leftrightarrow Q $
$ T $ $ T $ $ T $
$ T $ $ F $ $ F $
$ F $ $ T $ $ F $
$ F $ $ F $ $ T $



$ \underline { \rm DEFINITION } $ : Let $ P $ and $ Q $ be propositions. The following two theorems present a list of equivalent propositions. These equivalences will allow us to easily change from one propositional form to another.

$ \underline { \rm THEOREM \ 1.1 } $ : Let $ P $ and $ Q $ be propositions. $ \underline { \rm PROOF } $ : $ \underline { \rm EXAMPLE } $ : Equivalence of the following statements follows from Theorem 1.1 a.). $ \underline { \rm EXAMPLE } $ : The following statements are not equivalent. See Theorem 1.1 b.). $ \underline { \rm THEOREM \ 1.2 } $ : Let $ P, Q, $ and $ R $ be propositions. $ \underline { \rm PROOF } $ : All are easily proven using truth tables.

$ \underline { \rm NOTE } $ : Parts a.) and b.) in Theorem 1.2 make the connection between ``and" and ``or" statements. Part d.) makes the connection between ``if, then" and ``or" statements. Parts e.) and f.) represent associative properties for ``and" and ``or" statements. Parts g.) and h.) illustrate distributive properties.

Because the English language and use of common words can sometimes be ambiguous, we make the following mention of equivalent statements.

$ \underline { \rm NOTE } $ : On The Proof Page we will assume that the statement $ P \Rightarrow Q $ (If $ P $, then $ Q $) is equivalent to EACH of the following statements.
$ \underline { \rm EXAMPLE } $ : The following sets of statements are equivalent.
$ \underline { \rm NOTE } $ : On The Proof Page we will assume that the statement $ \sim Q \Rightarrow P $ (If not $ Q $, then $ P $) is equivalent to EACH of the following statements.
$ \underline { \rm EXAMPLE } $ : The following four statements are equivalent.
$ \underline { \rm EXAMPLE } $ : Rewrite each of the following sentences in symbolic propositional form. Then write each sentence in conditional (If ..., then ...) form. Find solutions HERE .
$ \underline { \rm EXAMPLE } $ : Rewrite each of the following sentences in symbolic propositional form. Then use a truth table to prove that they are equivalent. Find solutions HERE .
NOTE : (ORDER OF OPERATIONS) If parantheses are not used to clearly indicate the use of connectives, then invoke the connectives in the following order- $ \sim $ , $ \wedge $ , $ \vee $, $ \Rightarrow $, $ \Leftrightarrow $. For example, $ P \Rightarrow Q \wedge R \Leftrightarrow S $ is equivalent to $ (P \Rightarrow (Q \wedge R)) \Leftrightarrow S $ .

$ \underline { \rm EXAMPLE } $ : Provide parantheses to clarify the following ambiguous expressions. Find solutions HERE .




RETURN to The Proof Page .





Please e-mail your comments, questions, or suggestions to D. A. Kouba at kouba@math.ucdavis.edu .



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Duane Kouba 2002-05-11