Ehrenfeucht–Fraïssé game
Do you remember playing "spot the difference" games? Where you had two pictures, and you had to find the things that were different between them? The Ehrenfeucht-Fraïssé game is like that, but with math!
So, let's say we have two sets of numbers, set A and set B. We want to compare them and see if they're the same or if one is bigger than the other.
But we don't just want to compare the numbers themselves, we want to compare how the numbers relate to each other. For example, let's say A = {1, 2, 3} and B = {4, 5, 6}. We can see that A and B have different numbers, but we need to compare them more closely.
That's where the game comes in. We have two players, Player 1 and Player 2. Player 1 chooses a number from set A and Player 2 chooses a number from set B. Let's say Player 1 chooses 1 and Player 2 chooses 4.
Now, we have to see if there's a way to "map" all the other numbers in set A to numbers in set B so that the relationship between the two sets is the same as the relationship between 1 and 4. This means that if we can find a way to match up all the numbers in the sets, then they're the same size and have the same relationship.
For example, we could map 2 in set A to 5 in set B, and 3 in set A to 6 in set B. Now, we can see that the relationship between the two sets is the same as the relationship between 1 and 4. This means that A and B are the same size and have the same relationship!
But what if we can't find a way to match up all the numbers? Then we know that one set is "bigger" than the other. For example, if set A = {1, 2, 3, 4} and set B = {5, 6}, no matter how we map the numbers, we'll always have at least one number in A that doesn't have a corresponding number in B.
And that's the Ehrenfeucht-Fraïssé game! It's a way to compare sets of numbers by seeing if they have the same size and relationship between the numbers.
So, let's say we have two sets of numbers, set A and set B. We want to compare them and see if they're the same or if one is bigger than the other.
But we don't just want to compare the numbers themselves, we want to compare how the numbers relate to each other. For example, let's say A = {1, 2, 3} and B = {4, 5, 6}. We can see that A and B have different numbers, but we need to compare them more closely.
That's where the game comes in. We have two players, Player 1 and Player 2. Player 1 chooses a number from set A and Player 2 chooses a number from set B. Let's say Player 1 chooses 1 and Player 2 chooses 4.
Now, we have to see if there's a way to "map" all the other numbers in set A to numbers in set B so that the relationship between the two sets is the same as the relationship between 1 and 4. This means that if we can find a way to match up all the numbers in the sets, then they're the same size and have the same relationship.
For example, we could map 2 in set A to 5 in set B, and 3 in set A to 6 in set B. Now, we can see that the relationship between the two sets is the same as the relationship between 1 and 4. This means that A and B are the same size and have the same relationship!
But what if we can't find a way to match up all the numbers? Then we know that one set is "bigger" than the other. For example, if set A = {1, 2, 3, 4} and set B = {5, 6}, no matter how we map the numbers, we'll always have at least one number in A that doesn't have a corresponding number in B.
And that's the Ehrenfeucht-Fraïssé game! It's a way to compare sets of numbers by seeing if they have the same size and relationship between the numbers.