Question

1. for the three - part question that follows, provide your answer to each question in the given workspace. identify each part with a coordinating response. be sure to clearly label each part of your response as part a, part b, and part c.

use the information below for part a, part b, and part c.
statement: ben runs track and plays football.
negation: ben does not run track, and he does not play football.

part a: the negation above is incorrect. identify the mistake.

part b: correctly write the negation of the statement provided above.

part c: write a complete sentence to explain how you negated the statement.

Explanation:

Part A

The original statement is a conjunction ("and" statement: \( P \land Q \), where \( P \) is "Ben runs track" and \( Q \) is "Ben plays football"). The negation of a conjunction \(
eg (P \land Q) \) is \(
eg P \lor
eg Q \) (by De Morgan's Laws), not \(
eg P \land
eg Q \). The mistake is using "and" instead of "or" in the negation and incorrectly negating both parts with "and" instead of applying De Morgan's Law.

Using De Morgan's Law for the negation of a conjunction \(
eg (P \land Q)=
eg P \lor
eg Q \), where \( P \) is "Ben runs track" and \( Q \) is "Ben plays football", the correct negation is "Ben does not run track, or he does not play football".

The original statement is a conjunction (using "and") of two propositions. By De Morgan's Laws, the negation of a conjunction \( P \land Q \) is the disjunction (using "or") of the negations of each proposition, so we change "and" to "or" when negating the conjunction and negate each part appropriately.

Answer:

The mistake is that for the conjunction "Ben runs track and plays football" ( \( P \land Q \) ), its negation should follow De Morgan's Law (\(
eg (P \land Q)=
eg P \lor
eg Q \)), but the given negation uses "and" (\(
eg P \land
eg Q \)) instead of "or" and misapplies the negation of a conjunction.

Part B