document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176997>Question 1176997</A>: <I><SPAN STYLE=\"word-break: break-all;\"> ~F → G  I  -  F v G<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850541 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=850541\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this logical expression and determine the relationship between `~F → G` and `F v G`.\r<BR>\n" );
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document.write( "**Understanding the Symbols**\r<BR>\n" );
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document.write( "* `~`: Negation (NOT)<BR>\n" );
document.write( "* `→`: Implication (IF...THEN)<BR>\n" );
document.write( "* `v`: Disjunction (OR)\r<BR>\n" );
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document.write( "**Breaking Down the Expressions**\r<BR>\n" );
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document.write( "1.  **`~F → G`** (NOT F implies G)<BR>\n" );
document.write( "    * This statement is true in all cases except when `~F` is true and `G` is false.<BR>\n" );
document.write( "    * In other words, it's false only when `F` is false and `G` is false.\r<BR>\n" );
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document.write( "2.  **`F v G`** (F OR G)<BR>\n" );
document.write( "    * This statement is true when either `F` is true, `G` is true, or both are true.<BR>\n" );
document.write( "    * It is only false when both `F` and `G` are false.\r<BR>\n" );
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document.write( "**Truth Table**\r<BR>\n" );
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document.write( "To see the relationship clearly, let's create a truth table:\r<BR>\n" );
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document.write( "| F     | G     | ~F    | ~F → G | F v G |<BR>\n" );
document.write( "| :---- | :---- | :---- | :----- | :---- |<BR>\n" );
document.write( "| True  | True  | False | True   | True  |<BR>\n" );
document.write( "| True  | False | False | True   | True  |<BR>\n" );
document.write( "| False | True  | True  | True   | True  |<BR>\n" );
document.write( "| False | False | True  | False  | False |\r<BR>\n" );
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document.write( "**Analysis**\r<BR>\n" );
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document.write( "* Notice that the truth values for `~F → G` and `F v G` are identical in all rows of the truth table.\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "Therefore, `~F → G` is logically equivalent to `F v G`. They have the same truth values under all possible combinations of `F` and `G`.<BR>\n" );
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