document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167106>Question 1167106</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I would like to know how can I solve below.\r<BR>\n" );
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document.write( "F / (G->H) v (~G->J) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852341 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=852341\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem involves evaluating the truth value of the compound expression $F \lor (G \to H) \lor (\sim G \to J)$.\r<BR>\n" );
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document.write( "The key to solving this is understanding the truth tables for **disjunction** ($\lor$), **implication** ($\to$), and **negation** ($\sim$).\r<BR>\n" );
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document.write( "## Solving the Expression\r<BR>\n" );
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document.write( "The expression you provided is:<BR>\n" );
document.write( "$$F \lor (G \to H) \lor (\sim G \to J)$$\r<BR>\n" );
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document.write( "### 1. The Disjunction Rule ($\lor$)\r<BR>\n" );
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document.write( "The entire expression is a large **disjunction** (OR statement) connecting three main clauses:<BR>\n" );
document.write( "$$\text{CLAUSE}_1 \lor \text{CLAUSE}_2 \lor \text{CLAUSE}_3$$\r<BR>\n" );
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document.write( "The fundamental rule for disjunction is:<BR>\n" );
document.write( "$$\text{A disjunction is **TRUE** if at least one of its components is **TRUE**.} \quad (T \lor X = T)$$\r<BR>\n" );
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document.write( "Therefore, if **any** of the three main clauses is true, the entire expression is **TRUE**, regardless of the truth values of the other clauses.\r<BR>\n" );
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document.write( "### 2. The Implication Rule ($\to$)\r<BR>\n" );
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document.write( "The two complex clauses use the **implication** (IF...THEN) operator.<BR>\n" );
document.write( "$$\text{A} \to \text{B} \text{ is **FALSE** only when the antecedent (A) is **TRUE** and the consequent (B) is **FALSE**.} \quad (T \to F = F)$$\r<BR>\n" );
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document.write( "For all other combinations, the implication is **TRUE**.\r<BR>\n" );
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document.write( "### 3. Analysis by Case\r<BR>\n" );
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document.write( "To see if the entire expression can ever be false, we must check if it's possible for **all three clauses to be false simultaneously.**\r<BR>\n" );
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document.write( "$$\underbrace{F}_{\text{Clause 1}} \lor \underbrace{(G \to H)}_{\text{Clause 2}} \lor \underbrace{(\sim G \to J)}_{\text{Clause 3}} = F$$\r<BR>\n" );
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document.write( "This requires:<BR>\n" );
document.write( "1.  **Clause 1 is False:** $F$ must be **False**.<BR>\n" );
document.write( "    $$F = F$$\r<BR>\n" );
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document.write( "2.  **Clause 2 is False:** $G \to H$ must be **False**.<BR>\n" );
document.write( "    This only happens when the antecedent is True and the consequent is False:<BR>\n" );
document.write( "    $$G = T \quad \text{and} \quad H = F$$\r<BR>\n" );
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document.write( "3.  **Clause 3 is False:** $\sim G \to J$ must be **False**.<BR>\n" );
document.write( "    This only happens when the antecedent ($\sim G$) is True and the consequent ($J$) is False:<BR>\n" );
document.write( "    $$\sim G = T \quad \text{and} \quad J = F$$\r<BR>\n" );
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document.write( "### 4. Checking for Consistency\r<BR>\n" );
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document.write( "Let's combine the requirements from all three clauses:\r<BR>\n" );
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document.write( "* From Clause 1: $F = F$<BR>\n" );
document.write( "* From Clause 2: $G = T$<BR>\n" );
document.write( "* From Clause 2: $H = F$<BR>\n" );
document.write( "* From Clause 3: $J = F$<BR>\n" );
document.write( "* From Clause 3: $\sim G = T$\r<BR>\n" );
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document.write( "We have a **contradiction** because one condition requires $G = T$ and another requires $\sim G = T$ (which means $G = F$). It is logically impossible for both $G$ to be True and $G$ to be False simultaneously.\r<BR>\n" );
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document.write( "Since it is **impossible** for all three clauses to be false, the entire expression must always be **TRUE**.\r<BR>\n" );
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document.write( "## Conclusion\r<BR>\n" );
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document.write( "The expression $F \lor (G \to H) \lor (\sim G \to J)$ is a **tautology**; it is **always True** for any combination of truth values for $F$, $G$, $H$, and $J$. </SPAN>\n" );
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