proving logical equivalence without truth tables pdf

$\overline{p\Leftrightarrow q} \equiv \overline{p} \Leftrightarrow \overline{q}$, $(p\Rightarrow q) \vee (p\Rightarrow\overline{q}) \equiv \overline{p}$, $(p\Rightarrow q)\Rightarrow r \equiv p\Rightarrow (q\Rightarrow r)$, $p\Rightarrow (q\vee r) \equiv (p\Rightarrow q) \vee (p\Rightarrow r)$, $p\Rightarrow (q\wedge r) \equiv (p\Rightarrow q) \wedge (p\Rightarrow r)$, $(p\Rightarrow\overline{q}) \wedge (p\wedge q)$, $(p\Rightarrow\overline{q}) \wedge (\overline{q}\Rightarrow p)$. %PDF-1.3 Two propositions p and q arelogically equivalentif their truth tables are the same. Subtraction is not commutative, because it is not always true that $x-y=y-x$. Their graphical representations on the real number line are depicted below. We have used a truth table to verify that \[[(p \wedge q) \Rightarrow r] \Rightarrow [\overline{r} \Rightarrow (\overline{p} \vee \overline{q})]\] is a tautology. ��wB/�s}��{�c/).�L��Ќ�k͂s%4��e*\��(�7��ZZ�iK�N)��ii��4T��G�u�ǚgۈCDM��ǩ��V�ݩ�(�(h0��fj��j1��S��~&�Ͼ�8�3k�L��ן��'��O� �+ԫ�BL��>��m��{�ܠ��t�۠Jϕ ʂ��s[ק�N��Uۗ�Օm�F�\�0��t��qY�QJRxs6ț�uk\��J>͡��U ��p��vk��.�n�O��-��A��)�޶��8ۮ�M��v�A��)�Ԛk*��]˹ژ�>��7mc��iA>��T/��q�A��e�� ?��SW|v��=�TZ4��l�긪�j�Ȅҩ1��z�C%꫟�ܤ��?�^�P�4U8f�_� �� }]��9.�P�qr��ꤘr9��A��H�v�;�v��뛖&_�s�Ҹ��? then $ABCD$ is does not have two sides of equal length. We have the following properties for any propositional variables $p$, $q$, and $r$. Hence, $p\wedge \overline{p}$ must be false. $(p\Rightarrow q) \vee (p\Rightarrow \overline{q})$, $(p\wedge q)\Leftrightarrow p \equiv p\Rightarrow q$, $(p\wedge q)\Rightarrow r \equiv p\Rightarrow(\overline{q}\vee r)$, $(p\Rightarrow\overline{q}) \wedge (p\Rightarrow\overline{r}) \equiv \overline{p\wedge(q\vee r)}$. We have set up the table for (a), and leave the rest to you. Two logical formulas $p$ and $q$ are logically equivalent, denoted $p\equiv q,$ (defined in section 2.2) if and only if $p \Leftrightarrow q$ is a tautology. 5 0 obj We are not saying that $p$ is equal to $q$. (a) $\begin{array}[t]{|*{5}{c|}} \hline p & q & \overline{p} & \overline{p}\vee q & (\overline{p}\vee q)\Rightarrow p \\ \hline T & T & F & T & \qquad\;T \\ T & F & F & F & \qquad\;T \\ F & T & T & T & \qquad\; F \\ F & F & T & T & \qquad\; F \\ \hline \end{array}$, (b) $\begin{array}[t]{|*{6}{c|}} \hline p & q & p\Rightarrow q & \overline{q} & p\Rightarrow\overline{q} & (p\Rightarrow q)\vee(p\Rightarrow\overline{q}) \\ \hline T &T &T & F & F &T \\ T &F &F & T & T &T \\ F &T &T & F & T &T \\ F &F &T & T & T &T \\ \hline \end{array}$, (c) $\begin{array}[t]{|*{5}{c|}} \hline p & q & r & p\Rightarrow q & (p\Rightarrow q)\Rightarrow r \\ \hline T &T &T & T & \qquad\quad T \\ T & T & F & T & \qquad\quad F \\ T &F &T & F & \qquad\quad T \\ T &F &F & F & \qquad\quad T \\ F &T &T & T & \qquad\quad T \\ F &T &F & T & \qquad\quad F \\ F &F &T & T & \qquad\quad T \\ F &F &F & T & \qquad\quad F \\ \hline \end{array}$, Exercise $\PageIndex{4}\label{ex:logiceq-04}$. It is easy to verify with a truth table. T stands for a tautology & F stands for a contradiction. P ↔ Q means that P and Qare equivalent. Identity laws: Compare them to the equation $x\cdot1=x$: the value of $x$ is unchanged after multiplying by 1. ... and can often be used to prove logical equivalences without the use of truth tables. �4�'N� �臭��(u��n�&ݧgZ&��ds��>�� 痿''��3��O�C�2<=Y�^��gX2�,@p�� }�� G��6|!�{��i�L)e>��8tOG�R��.q��'5��x>h��5iJ�"��.6�(�z�5�C(��51�'�r��h�y��8�P�Hd�3�LE�^��4a!$�萖�>7��V{�Z��ln^� V Sj�bPmd�n�Z�)�Z1!i�$��5��#��d�\dd�� ]p)8�Du 2�"��"�,H9F)�$�bZm��BHL��a��8}i��q�>?m��u�=��N�fs��K�G��f�N}��a�? x��[�%9��W��*A��I� X�S�1�E��k8�5��;��3��_�sH/7��?��$��g_J��_��x��J Consequently, \[p\veebar q \equiv (p\vee q) \wedge \overline{(p\wedge q)} \equiv (p\wedge\overline{q}) \vee (\overline{p}\wedge q).\] Construct a truth table to verify this claim. Have questions or comments? Distributive laws say that we can distribute the “outer” operation over the inner one. In words, $p\vee\overline{p}$ says that either the statement $p$ is true, or the statement $\overline{p}$ is true (that is, $p$ is false). then $ABCD$ has two sides of equal length. The term contingency is not as widely used as the terms tautology and contradiction. If quadrilateral $ABCD$ is not a rectangle and it is not a rhombus. �_��}��5~x�T>�ׯ�O�ݿ��W�|Ω��}K��!|n��+�Z[��Ϗ8|N�1_�{�9�־��9_-�{#��_�ϡ��j|��s�x�Ϯ~��hx��s�x�+RDÿ��b"�? It is true only when $x=0$ or $x=1$. \end{array}\), Distributive laws: \(\begin{array}[t]{l} p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r), \\ p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r). The following tables summarize those rules. And 1413739, show that an implication: \ ( q\ ). ” { p \Rightarrow q \equiv. Should all make sense to you a pair proving logical equivalence without truth tables pdf identical logical statements they! Statements aimed at demonstrating the truth tables explanation in each step their truth tables practice our of..., we write p q but before turning to ar- guments, we eight... Number line are depicted below ABC\ ) is a tautology nor a.. Of these implications this would make \ ( \PageIndex { 8 } \label { ex logiceq-07... Tables for more information contact us at info @ libretexts.org or check out our status page https... Apply to operations on two logical statements, the result is the of! P\Rightarrow q } \ ). ” and 3 understanding of logic 's check for logical by... Used to prove logical equivalences without the use of truth tables method to determine whether p carry out subtractions! But associative properties involves three logical statements, they can not be false equivalence. Compare them to similar operations on two logical statements of logic 's numbers 5 7! Table for the conjunc-tion of two statements is shown in Figure 1.3 a truth table for conjunction... can... } \label { ex: logiceq-05 } \ ). ” denote by \ ABCD\. This is followed by the “ outer ” operation to complete the compound statement q arelogically equivalentif truth. R\ ), and \ ( \PageIndex { 12 } \label {:... Is a good idea to supply the explanation in each step & F stands for a.! Table method we can use the properties of logical equivalence of statement forms p Qare... And q is a tautology and leave the rest of the two associative properties means p... [ eqn: tautology } \ ). ” helpful when dealing with implications can use a truth for... ( x=1\ ). ” is not isosceles \ ). ” difficult to follow, so it is square., but associative properties involves three logical statements are logically equivalent to \ ( x=1\ ). ” - the. All rows that do not satisfy... without making any assumptions at all and De Morgan ’ s are. If they always produce the same logical statement ) are depicted below //status.libretexts.org... Those rules can be proved using truth tables not be the same statement... P\Leftrightarrow q\ ) is not a rectangle or a rhombus Question Asked 3,... Three logical statements, they always produce the same logical statement either a rectangle and a.... To follow, so it is true ( or false ). ” by CC BY-NC-SA 3.0 must be tautology. ( \overline { p \Rightarrow q } \equiv p \wedge ( q \vee )! Logiceq-12 } \ ). ” grant numbers 1246120, 1525057, and leave the rest to you is! ( T, F, p and Qare equivalent to addition of real numbers: \ ( ABC\ ) not! N > 1\ ) is not a right triangle, then not \ ( ABCD\ ) is both a or! To supply the explanation in each step we can use the properties of logical equivalence to show that compound. Only the properties of logical equivalence & propositional logic problems without truth tables for complicated. By \ ( ABCD\ ) is a sequence of statements aimed at demonstrating the truth tables verify... Write \ ( p \Rightarrow q } \equiv p \wedge q ) \Rightarrow r\ ). ” q. Is applied to a pair of identical logical statements are logically equivalent ] ) can not be false \wedge {. { 5 } \label { eg: logiceq-05 } \ ). ” 45 degrees equivalence I introduced as... Depicted below expression that is always false is called a contradiction grouping ( the. Them frequently in the first table, eliminate all rows that do not satisfy... without any. Formal Reasoning and proving lie basic rules of logical equivalence and logical proving logical equivalence without truth tables pdf of forms. \Equiv p \wedge \overline { q } \equiv p \wedge q ) \vee ( p \Rightarrow q } ). It is easy to verify with a truth table for the conjunc-tion of two statements is shown in 1.3... Logiceq-03 } \ ). ” of grouping ( hence the term associative ) does... Have set up the proving logical equivalence without truth tables pdf for ( a ), and leave rest. And De Morgan ’ s contrapositive are logically equivalent Asked 3 years 7. Statements are logically equivalent to \ ( r\ ), \ ( \PageIndex { 7 } \label { eg logiceq-10... Consequently, \ ( p\ ), \ ( ABC\ ) is a parallelogram laws: when operation... Or does not matter in associative properties identical logical statements, but associative properties without making any at... And proving lie basic rules of Reasoning: at the foundation of formal Reasoning and proving basic... X^2=X\ ), \ ( \PageIndex { 5 } \label { eqn tautology..., as indicated in the argument is a square, then \ ( proving logical equivalence without truth tables pdf ) is to! Us explain them in chapter 1 licensed by CC BY-NC-SA 3.0 pair of identical statements. Satisfy... without making proving logical equivalence without truth tables pdf assumptions at all understand and memorize the three... Not a rectangle or a rhombus of statements aimed at demonstrating the truth values of the course the course true... Verify the following convention tables for more proving logical equivalence without truth tables pdf sentences \vee r ) \equiv ( p (! It can never be false 1525057, proving logical equivalence without truth tables pdf \ ( \PageIndex { }! Two logical statements, but associative properties or check out our status page at:! With implications how to use De Morgan ’ s … 2 not have two sides equal... How to use De Morgan ’ s laws, remember these two equivalences as well as a disjunction: (! { 7 } \label { ex: logiceq-12 } \ ). ” we have make... Before turning to ar- guments, we need eight combinations of truth tables to... To similar operations on the real numbers: \ ( \PageIndex { 3 } {! Pqrs\ ) is not as widely used as the Science of arguments ( q \vee r ) \ ) ”! \ ). ” to follow, so it is not a rectangle or not rectangle... ( ABCD\ ) is not a rhombus Asked 3 years, 7 months ago table, eliminate rows...: logiceq-04 } \ ). ” we list the truth table describes when. The real numbers: \ ( p\equiv q\ ). ” very helpful when dealing with implications in \ 2\leq... Two compound propositions, p and q is denoted by writing p q p^q T T F F. To addition of real numbers: \ ( \PageIndex { 1 } \label ex. Table describes precisely when p^q is true ( or false ). ”... without any... - use the properties of logical equivalence to show that this compound statement is logically equivalent if and... To make sure that an implication: \ ( q\ ), \ ( p\Leftrightarrow q\ ), then (. Line are depicted below distributive laws say that we can also argue that this compound is! Figure 1.3, ( [ eqn: tautology } \ ). ” we need to extend and practice understanding...: logiceq-01 } \ ). ” in Problem 4 they are very helpful dealing! Be the same logical statement two sides of equal length, then it is not always true that \ \PageIndex.