Question

write the conditional and converse for the statement. determine the truth values of the conditionals and converses. if false, find a counterexample. write a true biconditional statement if possible.
lauras soccer team plays on saturdays.
conditional: if △ select choice △, then △ select choice △.
converse: if △ select choice △, then △ select choice △.
counterexample: lauras soccer team △ select choice △. the conditional is △ select choice △. converse: if lauras soccer team is △ select choice △, then it is △ select choice △. because the converse is △ select choice △, a true biconditional statement cannot be written.

Explanation:

Step1: Identify the original statement

The original statement is "Laura’s soccer team plays on Saturdays."

Step2: Write the conditional statement

A conditional statement is in the form "If \( p \), then \( q \)". Here, \( p \) is "it is a soccer game of Laura’s team" (or more precisely, "Laura’s soccer team is playing") and \( q \) is "it is Saturday". So the conditional is: If Laura’s soccer team is playing, then it is Saturday. Wait, no, actually, the original fact is that the team plays on Saturdays. So the correct conditional is: If it is a day when Laura’s soccer team plays, then it is Saturday. Or more simply, If Laura’s soccer team plays, then it is Saturday. Wait, maybe better: The original statement is about when they play. So the conditional should be: If a day is when Laura’s soccer team plays, then the day is Saturday. But maybe more straightforward: If Laura’s soccer team plays, then it is Saturday.

Step3: Write the converse

The converse of "If \( p \), then \( q \)" is "If \( q \), then \( p \)". So the converse of the conditional (If Laura’s soccer team plays, then it is Saturday) is: If it is Saturday, then Laura’s soccer team plays.

Step4: Determine truth values

- **Conditional truth value**: Is the conditional true? The original statement says they play on Saturdays, but do they only play on Saturdays? The problem later mentions a counterexample: "Laura’s soccer team playing on Thursday". So if there's a day (Thursday) when they play, then the conditional "If Laura’s soccer team plays, then it is Saturday" is false, because we have a case where they play (on Thursday) and it's not Saturday. So the conditional is false, and the counterexample is Laura’s soccer team playing on Thursday (since that's a time they play but it's not Saturday).
- **Converse truth value**: The converse is "If it is Saturday, then Laura’s soccer team plays". Is this true? The original statement says they play on Saturdays, but does every Saturday they play? The problem doesn't say they play every Saturday, just that they play on Saturdays (maybe sometimes). Wait, but the counterexample for the conditional is about the team playing on a non - Saturday. For the converse, if there's a Saturday when they don't play, then the converse is false. But the problem's context: Let's see the given counterexample for the conditional is playing on Thursday. For the converse, suppose there's a Saturday when they don't play, but the problem doesn't mention that. Wait, the problem says "Because the converse is [true/false], a true biconditional statement cannot be written." Wait, let's re - evaluate.

Wait, the original statement: "Laura’s soccer team plays on Saturdays." So the conditional: If Laura’s soccer team plays, then it is Saturday. Is this true? If the team can play on other days (like Thursday, as the counterexample says), then the conditional is false (because playing on Thursday is a case where \( p \) (playing) is true and \( q \) (Saturday) is false).

The converse: If it is Saturday, then Laura’s soccer team plays. Is this true? If there are Saturdays when they don't play, then the converse is false. But the problem's counterexample for the conditional is playing on Thursday. Now, for the biconditional, a biconditional " \( p \) if and only if \( q \)" is true only if both the conditional and converse are true. Since the conditional is false (because of Thursday) and the converse may also be false (if there's a Saturday they don't play), but the problem says "Because the converse is [let's see], a true biconditional statement cannot be written." Wait, let's fill in the blanks:

- Conditional: If [Laura’s soccer team plays], then [it is Saturday]. The conditional is [false] (because of the counterexample [Laura’s soccer team playing on Thursday]).
- Converse: If [it is Saturday], then [Laura’s soccer team plays]. The converse is [false]? Wait, no. Wait, the original statement is that they play on Saturdays. So maybe the converse: If it is Saturday, then Laura’s soccer team plays. Is this true? The original statement says they play on Saturdays, but does every Saturday they play? The problem doesn't say they play every Saturday, just that they play on Saturdays. But the counterexample for the conditional is about playing on a non - Saturday. For the converse, suppose we assume that on Saturdays they do play (from the original statement), but maybe there are Saturdays when they don't. But the problem's key is that the conditional is false (because of Thursday), and the converse: Let's see, if the converse were true, then every Saturday they play, but the original statement only says they play on Saturdays (not that they play every Saturday). But the problem's later text says "Because the converse is [select choice], a true biconditional statement cannot be written." Wait, maybe the converse is also false? Or maybe the converse is true? Wait, no. Let's go back.

Original statement: "Laura’s soccer team plays on Saturdays."

Conditional: If Laura’s soccer team plays, then it is Saturday. Is this true? No, because maybe they play on other days (like Thursday, as the counterexample). So conditional is false, counterexample is Laura’s soccer team playing on Thursday.

Converse: If it is Saturday, then Laura’s soccer team plays. Is this true? The original statement says they play on Saturdays, but does that mean they play every Saturday? The problem doesn't say they play only on Saturdays, just that they play on Saturdays. So maybe the converse is false (if there's a Saturday when they don't play), but the problem's counterexample is for the conditional. Wait, the problem's text: "Because the converse is [select choice], a true biconditional statement cannot be written." A biconditional is true only if both conditional and converse are true. Since the conditional is false (due to Thursday), and even if the converse were true, the biconditional would be false. But let's fill in the blanks as per the problem's structure:

- Conditional: If [Laura’s soccer team plays], then [it is Saturday]. The conditional is [false]. Counterexample: [Laura’s soccer team playing on Thursday].
- Converse: If [it is Saturday], then [Laura’s soccer team plays]. The converse is [false]? Or maybe [true]? Wait, no. Let's think again. The original statement is "Laura’s soccer team plays on Saturdays." So the conditional is "If a day is when Laura’s soccer team plays, then the day is Saturday." The converse is "If a day is Saturday, then the day is when Laura’s soccer team plays."

Now, the conditional is false because there exists a day (Thursday) when they play and it's not Saturday. The converse: Is every Saturday a day when they play? The original statement says they play on Saturdays, but maybe not every Saturday. But the problem's counterexample for the conditional is playing on Thursday. For the converse, if we take the original statement as "they play on Saturdays" (i.e., at least some Saturdays), but the converse would require that all Saturdays are days they play. If there's a Saturday when they don't play, then the converse is false. But the problem doesn't mention that. However, the key point is that the conditional is false (because of Thursday), so the biconditional can't be written (since biconditional needs both conditional and converse to be true).

So to summarize:

- Conditional: If Laura’s soccer team plays, then it is Saturday. (False, counterexample: Laura’s soccer team playing on Thursday)
- Converse: If it is Saturday, then Laura’s soccer team plays. (Let's assume, from the problem's context, that maybe the converse is also false, or maybe the problem is structured so that the conditional is false (due to Thursday), and the converse: let's see, the problem says "Because the converse is [select choice], a true biconditional statement cannot be written." So if either the conditional or the converse is false, the biconditional is false. Since the conditional is false (because of Thursday), the biconditional can't be written.

Answer:

• Conditional: If Laura’s soccer team plays, then it is Saturday (False, Counterexample: Laura’s soccer team playing on Thursday)
• Converse: If it is Saturday, then Laura’s soccer team plays (Truth value: Depends, but since conditional is false, biconditional can't be written)

But following the problem's blanks:

• Conditional: If \(\boldsymbol{\text{Laura’s soccer team plays}}\), then \(\boldsymbol{\text{it is Saturday}}\). The conditional is \(\boldsymbol{\text{false}}\), Counterexample: \(\boldsymbol{\text{Laura’s soccer team playing on Thursday}}\)
• Converse: If \(\boldsymbol{\text{it is Saturday}}\), then \(\boldsymbol{\text{Laura’s soccer team plays}}\). The converse is \(\boldsymbol{\text{false}}\) (or maybe true? But since conditional is false, biconditional can't be written)

Final answer for the blanks (as per the problem's structure):

Conditional: If Laura’s soccer team plays, then it is Saturday (False, Counterexample: Laura’s soccer team playing on Thursday)

Converse: If it is Saturday, then Laura’s soccer team plays (False or True? But the key is that the conditional is false, so biconditional can't be written)

But to fill the select choices:

• Conditional: If [Laura’s soccer team plays], then [it is Saturday]. The conditional is [false], Counterexample: [Laura’s soccer team playing on Thursday]
• Converse: If [it is Saturday], then [Laura’s soccer team plays]. The converse is [false] (or maybe true, but the problem says biconditional can't be written because converse is [let's say false or because conditional is false])

But the main answer for the conditional and converse:

Conditional: If Laura’s soccer team plays, then it is Saturday (False, counterexample: Laura’s soccer team playing on Thursday)

Converse: If it is Saturday, then Laura’s soccer team plays (Truth value: If we assume from the original statement that they play on Saturdays, but not that they only play on Saturdays, the converse may be false if there are Saturdays they don't play, or true if they play every Saturday. But the problem's counterexample for the conditional is playing on Thursday, so the conditional is false, hence biconditional can't be written.