Let S be a set of three elements given by S = {A, B, C}. In the following table,
ID: 1889319 • Letter: L
Question
Let S be a set of three elements given by S = {A, B, C}. In the following table, all the elements of S are listed in a row at the top and in a column at the left. The result of x * y is found in the row that starts with x at the left and in the column that has y at the top. For example, B * C = C and C * B = A.* A B C
A C A B
B A B C
C B A C
Is the binary operation * commutative? Why?
Determine whether there is an identity element in S with respect to *.
If there is an identity element, which elements have inverses?
Explanation / Answer
Let S be a set of three elements given by S = {A, B, C}. In the following table, all the elements of S are listed in a row at the top and in a column at the left. The result of x * y is found in the row that starts with x at the left and in the column that has y at the top. For example, B * C = C and C * B = A.
* A B C
A C A B
B A B C
C B A C
Is the binary operation * commutative? Why?
"Commutative" simply means that you can "commute" the varibles to different parts of the equation and get the same result.
For example, regular multiplication is commutative. One example of this is 2 x 3 = 3 x 2 = 6.
So let's test the outcomes here.
Here, we have:
A * B = A and B * A = B
But this means that A * B B * A.
So the binary operation is NOT commutative.
Determine whether there is an identity element in S with respect to *.
What is an identity element? For example, on the natural numbers, "1" is an identity element because no matter what we multply 1 by, we always get that number
2 x 1 = 2
3 x 1 = 3
etc.
Here, that would be either A, B, or C. So, does one of the letters always succumb to the preferenaces of the other letter, just as 1 always succumbs to the 2, 3, etc.?
Let's see.
A * A = C
A * B = A
A * C = B
B * A = A
B * B = B
B * C = C
Plus, C * B = C
Therefore B is our identy element, because no matter what we multiply by B, we always get that same letter.
If there is an identity element, which elements have inverses?
All of them! Inverse is like 1/something. 1/2 is the inverse of 2.
Here, B/A is the inverse of A, B/B = B is the inverse of B, and B/C is the inverse of C. Why? To see why, think of B as our "1". :)
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