Let's break down the problem step by step as asked. --- ## **Step 1: Assign Literals to Statements** Let’s define: | Literal | Meaning | |---------|-----------------------------------| | D | There is an economic downturn | | F | There are fewer jobs | | R | John has a good resume | | G | John will get a good job | | P | There are plenty of jobs | Note: "There are plenty of jobs" is logically the negation of "there are fewer jobs," so \( P \equiv \neg F \). --- ## **Step 2: Translate Statements into Propositional Logic** **1. If there is an economic downturn, there will be fewer jobs.** \[ D \rightarrow F \] --- **2. If there are fewer jobs and John has a good resume, he will get a good job.** \[ (F \wedge R) \rightarrow G \] --- **3. If there are plenty of jobs, John will get a good job.** \[ (\neg F) \rightarrow G \] --- **4. John has a good resume.** \[ R \] --- **5. There is an economic downturn.** \[ D \] --- ## **Step 3: Use Inference Rules to Prove “John will get a good job” (G) Given All Statements** ### **We have:** 1. \( D \rightarrow F \) 2. \( (F \wedge R) \rightarrow G \) 3. \( (\neg F) \rightarrow G \) 4. \( R \) 5. \( D \) **Let’s derive G step by step:** #### **Step 3.1: From 5 and 1** Given \( D \) and \( D \rightarrow F \): - By Modus Ponens: \( F \) #### **Step 3.2: Now have F and 4** We have \( F \) and \( R \). Statement 2: \( (F \wedge R) \rightarrow G \) - Both \( F \) and \( R \) are true. - By Modus Ponens: \( G \) **Conclusion:** **John will get a good job.** --- ## **Step 4: If KB Did Not Contain Statement 5 (\( D \)), Convert the First Four Statements to CNF** We have: 1. \( D \rightarrow F \) 2. \( (F \wedge R) \rightarrow G \) 3. \( (\neg F) \rightarrow G \) 4. \( R \) Let’s convert each to CNF. --- ### **Step 4.1: \( D \rightarrow F \)** - Equivalent to \( \neg D \vee F \) (already in CNF). --- ### **Step 4.2: \( (F \wedge R) \rightarrow G \)** - Equivalent to \( \neg (F \wedge R) \vee G \) - \( (\neg F \vee \neg R \vee G) \) (CNF). --- ### **Step 4.3: \( (\neg F) \rightarrow G \)** - Equivalent to \( F \vee G \) (CNF). --- ### **Step 4.4: \( R \)** - Already in CNF. --- **Summing up, KB in CNF:** 1. \( \neg D \vee F \) 2. \( \neg F \vee \neg R \vee G \) 3. \( F \vee G \) 4. \( R \) --- ## **Step 5: Use Resolution Algorithm to Prove \( G \)** ### **Add Negation of Goal (\( \neg G \)) to KB** Add \( \neg G \) to the set. Now, set of clauses: 1. \( \neg D \vee F \) 2. \( \neg F \vee \neg R \vee G \) 3. \( F \vee G \) 4. \( R \) 5. \( \neg G \) Let’s attempt to derive a contradiction. --- ### **Resolution Steps** #### **Step 1: Use 5 (\( \neg G \)) with 2 (\( \neg F \vee \neg R \vee G \))** - \( \neg F \vee \neg R \vee G \) - \( \neg G \) - Resolving on \( G \): Gives: \( \neg F \vee \neg R \) --- #### **Step 2: Use Result with 4 (\( R \))** - \( \neg F \vee \neg R \) - \( R \) - Resolving on \( R \): Gives: \( \neg F \) --- #### **Step 3: Use 3 (\( F \vee G \)) and 5 (\( \neg G \))** - \( F \vee G \) - \( \neg G \) - Resolving on \( G \): Gives: \( F \) --- #### **Step 4: Now have \( F \) and \( \neg F \) from above** - \( F \) (from Step 3) - \( \neg F \) (from Step 2) **Contradiction!** --- ## **Conclusion** By resolution, the knowledge base (KB) WITHOUT Statement 5 (\( D \)) and WITH the negation of the goal leads to a contradiction. Thus, \( G \) (John will get a good job) is entailed by the KB. --- ## **Final Answers and Explanations** --- ### **1. Literals and Propositional Logic** | Statement | Propositional Logic | |---------------------------------------------------------------------------|-------------------------------| | If there is an economic downturn, there will be fewer jobs. | \( D \rightarrow F \) | | If there are fewer jobs and John has a good resume, he will get a good job| \( (F \wedge R) \rightarrow G \) | | If there are plenty of jobs, John will get a good job. | \( (\neg F) \rightarrow G \) | | John has a good resume. | \( R \) | | There is an economic downturn. | \( D \) | --- ### **2. Inference Rule Proof (With All Statements)** - \( D \), \( D \rightarrow F \) ⇒ \( F \) - \( F \), \( R \), \( (F \wedge R) \rightarrow G \) ⇒ \( G \) **John will get a good job.** --- ### **3. KB (First Four Statements) in CNF** 1. \( \neg D \vee F \) 2. \( \neg F \vee \neg R \vee G \) 3. \( F \vee G \) 4. \( R \) --- ### **4. Resolution Proof (Without Statement 5)** Add \( \neg G \) to KB. From resolution, derive \( F \) and \( \neg F \). Contradiction → \( G \) is proved. --- ## **Summary Table** | Step | Result | |-------------------|--------------------------------------------------------| | Literals | D, F, R, G | | Logic | \( D \rightarrow F \), \( (F \wedge R) \rightarrow G \), \( (\neg F) \rightarrow G \), \( R \), \( D \) | | Inference | \( D \rightarrow F \rightarrow G \) | | CNF | \( \neg D \vee F \), \( \neg F \vee \neg R \vee G \), \( F \vee G \), \( R \) | | Resolution Proof | Contradiction ⇒ \( G \) is entailed | | **Final Answer** | **John will get a good job.** |