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). The negation of a conjunction \(p \land q\) should be \(
eg p \lor
eg q\) (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 as a conjunction.

Let \(p\) be "Ben runs track" and \(q\) be "Ben plays football". The original statement is \(p \land q\). By De Morgan's law, \(
eg(p \land q)=
eg p \lor
eg q\). So we negate each part and change "and" to "or".

The original statement is a conjunction of two propositions. Using De Morgan's laws, the negation of a conjunction \(p \land q\) is the disjunction of the negations of each proposition, i.e., \(
eg p \lor
eg q\). So we negate each part ("runs track" to "does not run track", "plays football" to "does not play football") and connect them with "or" instead of "and".

Answer:

The mistake is that for the statement "Ben runs track and plays football" (a conjunction), its negation should use "or" (disjunction) instead of "and" (conjunction) when negating the two parts. The negation incorrectly used "and" to connect the two negated statements.

Part B