document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171517>Question 1171517</A>: <I><SPAN STYLE=\"word-break: break-all;\">  For any TFL sentences 𝛼 and 𝛽 that are logically equivalent (i.e., whose truth values agree on every valuation of their sentence letters), does the following entailment hold: \r<BR>\n" );
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document.write( "𝛽 → 𝛼, 𝛼 ∨ 𝛽 ⊨ 𝛼 ∧ 𝛽\r<BR>\n" );
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document.write( "Could someone help me with this problem?\r<BR>\n" );
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document.write( "Thank you!<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 #850896 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=850896\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this entailment step-by-step.\r<BR>\n" );
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document.write( "**Given:**\r<BR>\n" );
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document.write( "* α and β are logically equivalent. This means they have the same truth values under every valuation (interpretation) of their sentence letters.\r<BR>\n" );
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document.write( "**Entailment to Prove:**\r<BR>\n" );
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document.write( "* β → α, α ∨ β ⊨ α ∧ β\r<BR>\n" );
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document.write( "**Analysis:**\r<BR>\n" );
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document.write( "1.  **Logical Equivalence:**<BR>\n" );
document.write( "    * Since α and β are logically equivalent, we can say α ≡ β. This means that whenever α is true, β is true, and whenever α is false, β is false.\r<BR>\n" );
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document.write( "2.  **β → α:**<BR>\n" );
document.write( "    * Because α ≡ β, the implication β → α is always true. If β is true, α is true, and if β is false, α is false. Thus the implication is always true.\r<BR>\n" );
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document.write( "3.  **α ∨ β:**<BR>\n" );
document.write( "    * Since α ≡ β, α ∨ β will be true whenever either α or β (or both) are true. In fact, due to logical equivalence, α ∨ β is true when α is true, and also true when β is true.\r<BR>\n" );
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document.write( "4.  **α ∧ β:**<BR>\n" );
document.write( "    * We want to show that α ∧ β is true in all cases where β → α and α ∨ β are true.<BR>\n" );
document.write( "    * Since β → α is always true, we only need to consider α ∨ β.<BR>\n" );
document.write( "    * If α ∨ β is true, then at least one of α or β is true.<BR>\n" );
document.write( "    * Because α ≡ β, if one is true, the other is also true.<BR>\n" );
document.write( "    * Therefore, both α and β are true.<BR>\n" );
document.write( "    * Hence, α ∧ β is true.\r<BR>\n" );
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document.write( "5.  **Entailment:**<BR>\n" );
document.write( "    * We need to check if in every valuation where β → α and α ∨ β are true, α ∧ β is also true.<BR>\n" );
document.write( "    * β → α is always true.<BR>\n" );
document.write( "    * If α ∨ β is true, then both α and β are true (due to logical equivalence).<BR>\n" );
document.write( "    * Therefore, α ∧ β is true.<BR>\n" );
document.write( "    * Thus, the entailment holds.\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "Yes, the entailment β → α, α ∨ β ⊨ α ∧ β holds.<BR>\n" );
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