document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191241>Question 1191241</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please prove the following arguments (10 questions, 1 point each):<BR>\n" );
document.write( "(1) P ∨ P ⊢ P<BR>\n" );
document.write( "(2) P ⊢ (P → Q) → Q<BR>\n" );
document.write( "(3) ∼(P & Q), P ⊢ ∼ Q<BR>\n" );
document.write( "(4) P ⊢ (∼(Q → R) → ∼ P) → (∼ R → ∼ Q))(4)<BR>\n" );
document.write( "(5) (P ∨ Q) → R ⊢ (P → R) & (Q → R)(5)<BR>\n" );
document.write( "(6) ⊢ (P ∨ Q) → (Q ∨ P)(6)<BR>\n" );
document.write( "(7) ∼(P & ∼ Q) ⊢ P → Q<BR>\n" );
document.write( "(8) (P ∨ Q) ↔ P ⊢ Q → P<BR>\n" );
document.write( "(9) P ↔ Q, Q ↔ R ⊢ P ↔ R<BR>\n" );
document.write( "(10) ⊢ (P → Q) → (∼ Q → ∼ P) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849155 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=849155\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here are the proofs for each argument:\r<BR>\n" );
document.write( "\n" );
document.write( "**(1) P ∨ P ⊢ P**\r<BR>\n" );
document.write( "\n" );
document.write( "| Statement | Reason |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 1. P ∨ P | Premise |<BR>\n" );
document.write( "| 2. P | Idempotent Law |\r<BR>\n" );
document.write( "\n" );
document.write( "**(2) P ⊢ (P → Q) → Q**\r<BR>\n" );
document.write( "\n" );
document.write( "| Statement | Reason |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 1. P | Premise |<BR>\n" );
document.write( "| 2. P → Q | Assumption |<BR>\n" );
document.write( "| 3. Q | Modus Ponens (1, 2) |<BR>\n" );
document.write( "| 4. (P → Q) → Q | Conditional Proof (2-3) |\r<BR>\n" );
document.write( "\n" );
document.write( "**(3) ∼(P & Q), P ⊢ ∼Q**\r<BR>\n" );
document.write( "\n" );
document.write( "| Statement | Reason |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 1. ∼(P & Q) | Premise |<BR>\n" );
document.write( "| 2. P | Premise |<BR>\n" );
document.write( "| 3. ∼P ∨ ∼Q | De Morgan's Law (1) |<BR>\n" );
document.write( "| 4. ∼Q | Disjunctive Syllogism (2, 3) |\r<BR>\n" );
document.write( "\n" );
document.write( "**(4) P ⊢ (∼(Q → R) → ∼P) → (∼R → ∼Q)**\r<BR>\n" );
document.write( "\n" );
document.write( "| Statement | Reason |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 1. P | Premise |<BR>\n" );
document.write( "| 2. ∼(Q → R) → ∼P | Assumption |<BR>\n" );
document.write( "| 3. ∼P | Modus Ponens (1,2) |<BR>\n" );
document.write( "| 4. ∼(Q → R) | Implication Elimination (2,3) |<BR>\n" );
document.write( "| 5. ∼(∼Q ∨ R) | Implication Equivalence (4)|<BR>\n" );
document.write( "| 6. Q & ∼R | DeMorgan's Law (5) |<BR>\n" );
document.write( "| 7. ∼R | Simplification (6) |<BR>\n" );
document.write( "| 8. ∼R → ∼Q | Conditional Proof (7) |<BR>\n" );
document.write( "| 9. (∼(Q → R) → ∼P) → (∼R → ∼Q) | Conditional Proof (2-8) |\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "**(5) (P ∨ Q) → R ⊢ (P → R) & (Q → R)**\r<BR>\n" );
document.write( "\n" );
document.write( "| Statement | Reason |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 1. (P ∨ Q) → R | Premise |<BR>\n" );
document.write( "| 2. P | Assumption |<BR>\n" );
document.write( "| 3. P ∨ Q | Introduction (2) |<BR>\n" );
document.write( "| 4. R | Modus Ponens (1, 3) |<BR>\n" );
document.write( "| 5. P → R | Conditional Proof (2-4) |<BR>\n" );
document.write( "| 6. Q | Assumption |<BR>\n" );
document.write( "| 7. P ∨ Q | Introduction (6) |<BR>\n" );
document.write( "| 8. R | Modus Ponens (1, 7) |<BR>\n" );
document.write( "| 9. Q → R | Conditional Proof (6-8) |<BR>\n" );
document.write( "| 10. (P → R) & (Q → R) | Conjunction (5, 9) |\r<BR>\n" );
document.write( "\n" );
document.write( "**(6) ⊢ (P ∨ Q) → (Q ∨ P)**\r<BR>\n" );
document.write( "\n" );
document.write( "| Statement | Reason |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 1. P ∨ Q | Assumption |<BR>\n" );
document.write( "| 2. P | Sub-assumption |<BR>\n" );
document.write( "| 3. Q ∨ P | Introduction (2) |<BR>\n" );
document.write( "| 4. Q | Sub-assumption |<BR>\n" );
document.write( "| 5. Q ∨ P | Introduction (4) |<BR>\n" );
document.write( "| 6. (P ∨ Q) → (Q ∨ P) | Conditional Proof (1-5) |\r<BR>\n" );
document.write( "\n" );
document.write( "**(7) ∼(P & ∼Q) ⊢ P → Q**\r<BR>\n" );
document.write( "\n" );
document.write( "| Statement | Reason |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 1. ∼(P & ∼Q) | Premise |<BR>\n" );
document.write( "| 2. ∼P ∨ ∼∼Q | De Morgan's Law (1) |<BR>\n" );
document.write( "| 3. ∼P ∨ Q | Double Negation (2) |<BR>\n" );
document.write( "| 4. P → Q | Implication Equivalence (3) |\r<BR>\n" );
document.write( "\n" );
document.write( "**(8) (P ∨ Q) ↔ P ⊢ Q → P**\r<BR>\n" );
document.write( "\n" );
document.write( "| Statement | Reason |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 1. (P ∨ Q) ↔ P | Premise |<BR>\n" );
document.write( "| 2. (P ∨ Q) → P | Biconditional Elimination (1) |<BR>\n" );
document.write( "| 3. Q | Assumption |<BR>\n" );
document.write( "| 4. P ∨ Q | Introduction (3) |<BR>\n" );
document.write( "| 5. P | Modus Ponens (2, 4) |<BR>\n" );
document.write( "| 6. Q → P | Conditional Proof (3-5) |\r<BR>\n" );
document.write( "\n" );
document.write( "**(9) P ↔ Q, Q ↔ R ⊢ P ↔ R**\r<BR>\n" );
document.write( "\n" );
document.write( "| Statement | Reason |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 1. P ↔ Q | Premise |<BR>\n" );
document.write( "| 2. Q ↔ R | Premise |<BR>\n" );
document.write( "| 3. P → Q | Biconditional Elimination (1) |<BR>\n" );
document.write( "| 4. Q → R | Biconditional Elimination (2) |<BR>\n" );
document.write( "| 5. P → R | Hypothetical Syllogism (3, 4) |<BR>\n" );
document.write( "| 6. R → Q | Biconditional Elimination (2) |<BR>\n" );
document.write( "| 7. Q → P | Biconditional Elimination (1) |<BR>\n" );
document.write( "| 8. R → P | Hypothetical Syllogism (6, 7) |<BR>\n" );
document.write( "| 9. P ↔ R | Biconditional Introduction (5, 8) |\r<BR>\n" );
document.write( "\n" );
document.write( "**(10) ⊢ (P → Q) → (∼Q → ∼P)**\r<BR>\n" );
document.write( "\n" );
document.write( "| Statement | Reason |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 1. P → Q | Assumption |<BR>\n" );
document.write( "| 2. ∼Q | Assumption |<BR>\n" );
document.write( "| 3. ∼P | Modus Tollens (1, 2) |<BR>\n" );
document.write( "| 4. ∼Q → ∼P | Conditional Proof (2-3) |<BR>\n" );
document.write( "| 5. (P → Q) → (∼Q → ∼P) | Conditional Proof (1-4) |<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );