If proposition 'All pens are not pencils' is taken to be true, then which of the following propositions can be false?
All pens are pencils
To analyze the given proposition and determine which option can be false, we need to understand the statement and its implications in logical terms.
The given proposition is: "All pens are not pencils." This essentially means that not every pen is a pencil. Another way to interpret this is that at least one pen is not a pencil.
We have four options to evaluate:
- All pens are pencils
- Some pencils are pens
- No pen is pencil
- Some pens are pencils
Let's evaluate each option:
- All pens are pencils: This statement contradicts the given proposition. The proposition states that not all pens are pencils. Therefore, this option must be false.
- Some pencils are pens: This is possible even if not all pens are pencils. There's no contradiction here, so this statement can be true.
- No pen is pencil: This is a possible scenario under the given proposition. If no pen is a pencil, then certainly not all pens are pencils. Hence, this statement can also be true.
- Some pens are pencils: This is possible if not all pens are pencils, implying that some still could be. There is no contradiction, so this statement can also be true.
Based on this evaluation, the option that can be false while the original statement is true is:
- All pens are pencils
The correct answer is "All pens are pencils," as it directly contradicts the given proposition.
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