document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164609>Question 1164609</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the following logical expression, place the operations in the order in which you would apply, from first operation to last.\r<BR>\n" );
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document.write( "~p ∧ q → r </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854341 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854341\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Following the standard rules of **Logical Precedence**, we evaluate the operations from the most \"tightly bound\" operator to the \"main connective.\" \r<BR>\n" );
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document.write( "The hierarchy is as follows: <BR>\n" );
document.write( "1. **Negation** ($\sim$) <BR>\n" );
document.write( "2. **Conjunction** ($\land$) <BR>\n" );
document.write( "3. **Disjunction** ($\lor$) <BR>\n" );
document.write( "4. **Conditional** ($\to$)\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **Order of Application**\r<BR>\n" );
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document.write( "For the expression **$\sim p \land q \to r$**, the operations should be applied in this order:\r<BR>\n" );
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document.write( "1.  **First: Negation ($\sim p$)**<BR>\n" );
document.write( "    The \"NOT\" operator is applied only to the variable immediately following it. You must flip the truth value of $p$ first.<BR>\n" );
document.write( "2.  **Second: Conjunction ($\sim p \land q$)**<BR>\n" );
document.write( "    The \"AND\" operator takes precedence over the conditional. You evaluate the relationship between the result of Step 1 and $q$.<BR>\n" );
document.write( "3.  **Last: Conditional ($\dots \to r$)**<BR>\n" );
document.write( "    The \"IF...THEN\" operator is the main connective here. It evaluates the relationship between the entire left-hand side (the antecedent) and $r$ (the consequent).\r<BR>\n" );
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document.write( "### **Summary Table**<BR>\n" );
document.write( "| Order | Operation | Symbol | Resulting Grouping |<BR>\n" );
document.write( "| :--- | :--- | :--- | :--- |<BR>\n" );
document.write( "| **1st** | Negation | $\sim$ | $(\sim p)$ |<BR>\n" );
document.write( "| **2nd** | Conjunction | $\land$ | $(\sim p \land q)$ |<BR>\n" );
document.write( "| **3rd** | Conditional | $\to$ | $((\sim p \land q) \to r)$ |\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "**Wait, why not the arrow first?**<BR>\n" );
document.write( "In logic, the conditional ($\to$) is considered \"weaker\" than conjunction or negation. If you wanted the arrow to be evaluated first, the expression would have required parentheses like this: $\sim p \land (q \to r)$. Without them, we follow the hierarchy above.\r<BR>\n" );
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document.write( "Are you working on a truth table for this specific statement? </SPAN>\n" );
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