Rules of Inference

 $p\rightarrow q,\ p\ \$  implies  $q$  also written  $\left(p\rightarrow q\right)\wedge p\Longrightarrow q$  

• Modus Ponnens
  

• Modus Tollens
  

• Disjunctive Syllogism
  

• Hypothetical Syllogism
  

 $\left(p\rightarrow q\right),\ \sim q\$  implies  $\sim p$  

• Modus Ponens
  

• Modus Tollens
  

• Disjunctive Syllogism
  

• Simplification

 $\left(p\rightarrow q\right),\ \left(q\rightarrow r\right)\$  implies  $\left(p\rightarrow r\right)$  

• Modus Ponens
  

• Modus Tollens
  

• Distributive Syllogism
  

• Hypothetical Syllogism
  

 $\left(p\vee q\right),\ \sim p$  implies  $q$  

• Modus Ponens
  

• Disjunctive Syllogism
  

• Simplification

• Addition

"If A works hard, then either B or C will enjoy themselves". " If B enjoys himself then A will not work hard".

"If D enjoys himself, then C will not".

Therefore, " if A works hard, D will not enjoy himself". Formulate these sentences

• $A\rightarrow\left(B\wedge C\right),\ B\longrightarrow A,\ D\longrightarrow C$ $Therefore\ A\ \rightarrow D$

• $A\rightarrow\left(B\vee C\right),\ B\longrightarrow\sim A,\ D\longrightarrow\sim C\$ $Therefore\ \ A\rightarrow\sim D$

• None of the above
  

 $p\vee q\Longleftrightarrow q\vee p\$  and  $p\wedge q\Longleftrightarrow q\wedge p$  

• Commutative laws

• De Morgan’s laws

• Idempotent laws

• Absorption law

 $p\vee p\Longleftrightarrow p$  and  $p\wedge p\Longleftrightarrow p$  

• De Morgan’s laws

• modus ponens

• Associative laws

• Idempotent laws

which one of the following is an absorption law?

•  $p\wedge\left(p\vee q\right)\Longleftrightarrow p$  and  $p\vee\left(p\wedge q\right)\Longleftrightarrow p$  

•  $p\wedge p\Longleftrightarrow p$  and  $p\vee p\Longleftrightarrow p$  

•  $p\wedge q\Longleftrightarrow q\wedge p$  and  $p\vee q\Longleftrightarrow q\vee p$  

•  $p\vee\left(q\vee r\right)\Longleftrightarrow\left(p\vee q\right)\vee r$  and  $p\wedge\left(q\wedge r\right)\Longleftrightarrow\left(p\wedge q\right)\wedge r$  

 $\sim\left(p\vee q\right)\Longleftrightarrow\sim p\wedge\sim q$  and  $\sim\left(p\wedge q\right)\Longleftrightarrow\sim p\vee\sim q$  

• Distributive laws

• Associative laws

• De Morgan’s laws

• Absorption law

 $p\wedge\left(q\vee r\right)\Longleftrightarrow\left(p\wedge q\right)\vee\left(p\wedge r\right)$  and  $p\vee\left(q\wedge r\right)\Longleftrightarrow\left(p\vee q\right)\wedge\left(p\vee r\right)$  

• Associative laws

• Distributive laws

• Absorption law

• none of the above