Question

determine whether the conclusion was made by the law of detachment or the law of syllogism. if it’s an invalid conclusion, write \invalid\. 21. given: if the kicker makes the field goal, then the team will win the game. given: if the team wins the game, then they will move on to the playoffs. conclusion: if the kicker makes the field goal, then the team will move on to the playoffs. valid by: law of syllogism 22. given: if a number is a natural number, then it is a whole number. given: x is a whole number conclusion: x is a natural number valid by: law of syllogism © gina wilson (all things algebra), 2014

Explanation:

Problem 21

Step1: Recall Law of Syllogism

The Law of Syllogism states that if we have two conditional statements \( p
ightarrow q \) and \( q
ightarrow r \), we can conclude \( p
ightarrow r \).

Step2: Identify \( p \), \( q \), \( r \)

• Let \( p \): "the kicker makes the field goal"
• Let \( q \): "the team will win the game"
• Let \( r \): "they will move on to the playoffs"

The first given is \( p
ightarrow q \) ("If the kicker makes the field goal, then the team will win the game"). The second given is \( q
ightarrow r \) ("If the team wins the game, then they will move on to the playoffs"). The conclusion is \( p
ightarrow r \) ("If the kicker makes the field goal, then the team will move on to the playoffs"), which matches the Law of Syllogism.

Problem 22

Step1: Analyze the Conditionals

The first given is \( p
ightarrow q \) where \( p \): "a number is a natural number", \( q \): "it is a whole number". The second given is "x is a whole number" (which is \( q \) is true), and the conclusion is "x is a natural number" (which would be \( p \) is true).

Step2: Recall Validity of Reasoning

The Law of Detachment requires \( p
ightarrow q \) and \( p \) to conclude \( q \). The Law of Syllogism requires two conditionals \( p
ightarrow q \) and \( q
ightarrow r \). Here, we have \( p
ightarrow q \) and \( q \) (not \( p \) or another conditional \( q
ightarrow r \)). The conclusion is an invalid use of reasoning because we can't conclude \( p \) from \( q \) (the converse of \( p
ightarrow q \) is not necessarily true; e.g., 0 is a whole number but not a natural number). So the conclusion is invalid, not by Law of Syllogism (or Detachment correctly here). But for the question of what validates (or in this case, the original answer was wrong, but the process for 21 is correct as above). Wait, 22's conclusion is invalid. But for the "Valid by" part, 21 is valid by Law of Syllogism, 22 is invalid.

Answer:

(for 21):
Valid by: Law of Syllogism