Statements: Some pens are books.
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Statements:** Some pens are books. All books are papers. All papers are files.
Conclusions:
I. Some files are pens.
II. Some papers are pens.
III. Some files are books.
Which of the conclusions logically follow?
Show answer & explanation
Using set logic: Statement 1 (Some pens are books) means pens ∩ books ≠ ∅. Statement 2 (All books ⊆ papers) and Statement 3 (All papers ⊆ files) create the chain: books ⊆ papers ⊆ files. Conclusion II is valid: since some pens are books and all books are papers, some pens must be papers. Conclusion III is valid: since all books are papers and all papers are files, all books are files, so some files are books. Conclusion I is invalid: while some pens are books and books are in files, this does not guarantee the reverse—some files being pens requires pens to be a subset of files, which is not established.
Step-by-step Derivation:
Step 1: Map the logical relationships.
- Pens ∩ Books ≠ ∅ (some overlap)
- Books ⊆ Papers (all books are papers)
- Papers ⊆ Files (all papers are files)
Step 2: Check Conclusion I (Some files are pens).
- Chain: some pens ∈ books, all books ∈ papers, all papers ∈ files
- Therefore: some pens ∈ files
- But this is 'some pens are files', NOT 'some files are pens'
- Direction matters: we cannot reverse this implication
- Conclusion I is INVALID
Step 3: Check Conclusion II (Some papers are pens).
- Some pens are books (given)
- All books are papers (given)
- By transitivity: some pens are papers
- Therefore: some papers are pens (reverse is logically equivalent)
- Conclusion II is VALID
Step 4: Check Conclusion III (Some files are books).
- All books are papers (given)
- All papers are files (given)
- By transitivity: all books are files
- Therefore: some files are books (since all books ⊆ files)
- Conclusion III is VALID
Conclusion: Only II and III follow.