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# Truth tables - Card question Watch

1. Each of the four cards has a number on one side and a letter on the other.

G Y 8 4

Which cards do you need to turn over to check whether it is true that every card with a Y on one side has an 8 on the other?

How do I go about answering a question like this? Help is much appreciated
2. hmmmmm....I wonder, how is this related to truth tables?
3. (Original post by boromir9111)
hmmmmm....I wonder, how is this related to truth tables?
I assume that there is an arbitrary method using the a implies b truth table so that every possible statement concerning the cards can be examined and verified
4. (Original post by Ludivico)
I assume that there is an arbitrary method using the a implies b truth table so that every possible statement concerning the cards can be examined and verified
There are two numbers shown there, I assume it cannot be any other number then? So if it's a 8 it has to be either G or Y on the other side?
5. From logically standpoint, you would need to turn over at most 4 cards, any card that has an 8 on one side most be turned to see if it has a Y on the other side. The truth table could be represented with what cards you need to turn, with Is 8, Isn't 8 but is number, Is Y, Isn't Y but is Letter on the top representing side A, and the same four terms on the side representing side B, with is 8 or is Y (on top or side) being you turn, is 8 / is Y (on side AND top) representing true fact, is 8 / is Y (on side but NOT on top or opposite) representing false fact, anything else is no turn as false due to not 8 or Y.
6. (Original post by boromir9111)
There are two numbers shown there, I assume it cannot be any other number then? So if it's a 8 it has to be either G or Y on the other side?
That's not specified
7. You only need to turn over one card surely?

The question implies that every card with letter Y has an 8 on the other side, but it doesn't say that every card with the number 8 has a Y on the opposite side.
8. (Original post by rmhumphries)
From logically standpoint, you would need to turn over at most 4 cards, any card that has an 8 on one side most be turned to see if it has a Y on the other side. The truth table could be represented with what cards you need to turn, with Is 8, Isn't 8 but is number, Is Y, Isn't Y but is Letter on the top representing side A, and the same four terms on the side representing side B, with is 8 or is Y (on top or side) being you turn, is 8 / is Y (on side AND top) representing true fact, is 8 / is Y (on side but NOT on top or opposite) representing false fact, anything else is no turn as false due to not 8 or Y.
Thank you for the lengthy response. I see you've discussed it even more generally than I needed.

In the particular case of the letters and numbers I have given. Is it necessary to turn other three cards on the right?
9. (Original post by wow62)
You only need to turn over one card surely?

The question implies that every card with letter Y has an 8 on the other side, but it doesn't say that every card with the number 8 has a Y on the opposite side.
I don't think the question implies that every card with letter Y has an 8 on the other side, it could have any other number on the other side?
10. (Original post by wow62;2905
1868
)

You only need to turn over one card surely?

The question implies that every card with letter Y has an 8 on the other side, but it doesn't say that every card with the number 8 has a Y on the opposite side.

(Original post by Ludivico)
Thank you for the lengthy response. I see you've discussed it even more generally than I needed.

In the particular case of the letters and numbers I have given. Is it necessary to turn other three cards on the right?
If you have the exact cards "G Y 8 4" and they can have any letter or any number, you need to turn two cards, the 2nd card (Y) and the 3rd card (8) as the 2nd could have an eight on the other side, and the third could have a Y on the other side. You need to check the 8, as if it has a Y on the other side, it is a card that has a Y on one side, and an eight on the side side.

In regards to the first quoted person, while the 8 may not have an 8 on the other side, it might. It depends on if you want to know if you are checking there exists a card with Y on one side, and 8 on the other, or how many there are. You also have to take into account that the card with a Y on the front side (the side you see) may not have an eight on the other side, if it doesn't then the eight might have a Y on the other side. Hence, even the "if exists" statement may need two cards.
11. (Original post by rmhumphries)
If you have the exact cards "G Y 8 4" and they can have any letter or any number, you need to turn two cards, the 2nd card (Y) and the 3rd card (8) as the 2nd could have an eight on the other side, and the third could have a Y on the other side. You need to check the 8, as if it has a Y on the other side, it is a card that has a Y on one side, and an eight on the side side.

In regards to the first quoted person, while the 8 may not have an 8 on the other side, it might. It depends on if you want to know if you are checking there exists a card with Y on one side, and 8 on the other, or how many there are. You also have to take into account that the card with a Y on the front side (the side you see) may not have an eight on the other side, if it doesn't then the eight might have a Y on the other side. Hence, even the "if exists" statement may need two cards.
Surely the card with 4 on it has to be turned to check whether it has a Y on the other side?
12. (Original post by Ludivico)
Surely the card with 4 on it has to be turned to check whether it has a Y on the other side?
No, because even if it had a 'Y', you know it doesn't have an '8', as it only has two sides. You questions says if there is a 'Y', there must be an '8'.
13. (Original post by rmhumphries)
No, because even if it had a 'Y', you know it doesn't have an '8', as it only has two sides. You questions says if there is a 'Y', there must be an '8'.
It asks whether it's true that every card with a Y has an 8. If a Y was on the other side of the 4 then it wouldn't be true that every card with a Y has an 8 so we have to check whether there's a Y there. On the other hand, we don't have to check G because it doesn't have a Y on it?
14. (Original post by Ludivico)
It asks whether it's true that every card with a Y has an 8. If a Y was on the other side of the 4 then it wouldn't be true that every card with a Y has an 8 so we have to check whether there's a Y there. On the other hand, we don't have to check F because it doesn't have a Y on it?
No. If there is a 4 on one side, there can not be an 8 on one side, and a 'Y' on the other. You could check if the other side was a 'Y', but even if it was, then when you turned it over, it wouldn't be an '8', you know it is a '4', so no point checking.
15. (Original post by rmhumphries)
No. If there is a 4 on one side, there can not be an 8 on one side, and a 'Y' on the other. You could check if the other side was a 'Y', but even if it was, then when you turned it over, it wouldn't be an '8', you know it is a '4', so no point checking.
Lol, I don't think I'm getting it completely. 8 => Y is what we're trying to show? If so, then it makes more sense, I was just trying to get it totally via 'common sense' first. Thank you for the help
16. You are to show:

For each card that has a Y on one side has an 8 on the other side. I got this from: "It asks whether it's true that every card with a Y has an 8."

Each card only has two sides. So a side x (where x represents either side) must be either 8 or Y. If not, the card doesn't matter, and we move on to the next card.
If a card has a Y or an 8 on one side, we turn it over. If it had a Y on the front and an 8 on the back, or an 8 on the front and a Y on the back, the statement holds, we return true,
If a card has a Y or an 8 on one side, and when we turn it over, it does not have an 8 if it had a Y on the front, or a Y if it had an 8 on the front (basically the opposite of what is on the front out of (Y,8)), then we return false.

If there is one false (so one card does not follow the rule), the rule is broken. If all cards were discarded or held true, the rule holds.
17. (Original post by Ludivico)
...
(Original post by rmhumphries)
Y...

Beg to differ.

OP is asked to check whether "Y one side" --> "8 other side", and we can only see one side to start, and for each card one side has a letter and the other a number. We need to check if any card violates that implication.

We do not need to check the first card, as it has a G, so "Y one side" is false and hence the implication is true.

We need to check the second card, as it has a Y, to see if it is an 8 on the reverse; if not the implication is false.

We don't need to check the third card as it has an 8 on one side and the implication is therefore true for that card, regardless of whether there is a Y on the reverse or not.

We do need to check the fourth card as it has a 4. If there is a Y on the reverse the implication is false.

So we need to check the 2nd and 4th cards only.
18. (Original post by ghostwalker)
Beg to differ.

OP is asked to check whether "Y one side" --> "8 other side", and we can only see one side to start, and for each card one side has a letter and the other a number. We need to check if any card violates that implication.

We do not need to check the first card, as it has a G, so "Y one side" is false and hence the implication is true.

We need to check the second card, as it has a Y, to see if it is an 8 on the reverse; if not the implication is false.

We don't need to check the third card as it has an 8 on one side and the implication is therefore true for that card, regardless of whether there is a Y on the reverse or not.

We do need to check the fourth card as it has a 4. If there is a Y on the reverse the implication is false.

So we need to check the 2nd and 4th cards only.
Brilliant explanation as always. That's what I was getting at for the 4th card. The 3rd card I missed though.

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