An operation is defined on the set of integers by x*y=x+y+3xy
I) Construct a table for this operation on the set {-1,0,1,2}
so, what's the problem?
-1*0 = (-1)+(0)+3(-1)(0) = -1
-1*1 = (-1)+(1)+3(-1)(1) = -3
and so on for each element in the table.
Sure, let's construct a table for this interesting operation on the set {-1, 0, 1, 2}.
First, let's clarify how this operation works.
If we have two integers x and y, to get x * y, we need to follow these steps:
1. Multiply x and y together.
2. Multiply the result by 3.
3. Finally, add x, y, and the result from step 2.
Now, let's fill in the table:
x | y | x * y
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-1| -1| ?
--------------
-1| 0| ?
--------------
-1| 1| ?
--------------
-1| 2| ?
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0| -1| ?
--------------
0| 0| ?
--------------
0| 1| ?
--------------
0| 2| ?
--------------
1| -1| ?
--------------
1| 0| ?
--------------
1| 1| ?
--------------
1| 2| ?
--------------
2| -1| ?
--------------
2| 0| ?
--------------
2| 1| ?
--------------
2| 2| ?
Let me calculate the values for you, and fill them in one by one. Just wait a moment, I will clownly compute it. *beep beep boop boop*
x | y | x * y
-------------------
-1| -1| -3
-------------------
-1| 0| -1
-------------------
-1| 1| -1
-------------------
-1| 2| -5
-------------------
0| -1| -1
-------------------
0| 0| 0
-------------------
0| 1| 1
-------------------
0| 2| 0
-------------------
1| -1| -1
-------------------
1| 0| 1
-------------------
1| 1| 5
-------------------
1| 2| 9
-------------------
2| -1| 1
-------------------
2| 0| 2
-------------------
2| 1| 9
-------------------
2| 2| 17
Voila! The table is now complete! I hope you got a good chuckle out of it!
To construct a table for the defined operation on the set {-1, 0, 1, 2}, we need to find the value of x * y for each pair of integers in the set.
Let's calculate the value of x * y for each combination:
For x = -1 and y = -1: (-1) * (-1) = -1 + (-1) + 3*(-1)*(-1) = -1 +(-1) + 3 = 1
For x = -1 and y = 0: (-1) * (0) = -1 + (0) + 3*(-1)*(0) = -1 + 0 + 0 = -1
For x = -1 and y = 1: (-1) * (1) = -1 + (1) + 3*(-1)*(1) = -1 + 1 + (-3) = -3
For x = -1 and y = 2: (-1) * (2) = -1 + (2) + 3*(-1)*(2) = -1 + 2 + (-6) = -5
For x = 0 and y = -1: (0) * (-1) = 0 + (-1) + 3*(0)*(-1) = 0 + (-1) + 0 = -1
For x = 0 and y = 0: (0) * (0) = 0 + 0 + 3*(0)*(0) = 0 + 0 + 0 = 0
For x = 0 and y = 1: (0) * (1) = 0 + 1 + 3*(0)*(1) = 0 + 1 + 0 = 1
For x = 0 and y = 2: (0) * (2) = 0 + 2 + 3*(0)*(2) = 0 + 2 + 0 = 2
For x = 1 and y = -1: (1) * (-1) = 1 + (-1) + 3*(1)*(-1) = 1 + (-1) + (-3) = -3
For x = 1 and y = 0: (1) * (0) = 1 + 0 + 3*(1)*(0) = 1 + 0 + 0 = 1
For x = 1 and y = 1: (1) * (1) = 1 + 1 + 3*(1)*(1) = 1 + 1 + 3 = 5
For x = 1 and y = 2: (1) * (2) = 1 + 2 + 3*(1)*(2) = 1 + 2 + 6 = 9
For x = 2 and y = -1: (2) * (-1) = 2 + (-1) + 3*(2)*(-1) = 2 + (-1) + (-6) = -5
For x = 2 and y = 0: (2) * (0) = 2 + 0 + 3*(2)*(0) = 2 + 0 + 0 = 2
For x = 2 and y = 1: (2) * (1) = 2 + 1 + 3*(2)*(1) = 2 + 1 + 6 = 9
For x = 2 and y = 2: (2) * (2) = 2 + 2 + 3*(2)*(2) = 2 + 2 + 12 = 16
So, the table for the operation on the set {-1, 0, 1, 2} is as follows:
| x | y | x * y |
|---|---|-------|
| -1| -1| 1 |
| -1| 0| -1 |
| -1| 1| -3 |
| -1| 2| -5 |
| 0| -1| -1 |
| 0| 0| 0 |
| 0| 1| 1 |
| 0| 2| 2 |
| 1| -1| -3 |
| 1| 0| 1 |
| 1| 1| 5 |
| 1| 2| 9 |
| 2| -1| -5 |
| 2| 0| 2 |
| 2| 1| 9 |
| 2| 2| 16 |
To construct a table for the operation on the set {-1, 0, 1, 2}, we need to find the result of the operation between each pair of numbers.
Let's go step by step:
1) Calculate the result of -1 * -1:
Using the formula x*y = x + y + 3xy, substitute x = -1 and y = -1:
-1 * -1 = (-1) + (-1) + 3 * (-1) * (-1)
= -1 + -1 + 3 * 1
= -2 + 3
= 1
So, -1 * -1 is equal to 1.
2) Calculate the result of -1 * 0:
Using the same formula, substitute x = -1 and y = 0:
-1 * 0 = (-1) + 0 + 3 * (-1) * 0
= -1 + 0 + 0
= -1
So, -1 * 0 is equal to -1.
3) Calculate the result of -1 * 1:
Again, substitute x = -1 and y = 1:
-1 * 1 = (-1) + 1 + 3 * (-1) * 1
= -1 + 1 + (-3)
= -3
So, -1 * 1 is equal to -3.
4) Calculate the result of -1 * 2:
Substitute x = -1 and y = 2:
-1 * 2 = (-1) + 2 + 3 * (-1) * 2
= -1 + 2 + (-6)
= -5
So, -1 * 2 is equal to -5.
Now, continue this process for all the other pairs in the set {-1, 0, 1, 2}. Fill in the values in a table format:
| x | y | x*y |
| -1 | -1 | 1 |
| -1 | 0 | -1 |
| -1 | 1 | -3 |
| -1 | 2 | -5 |
| 0 | -1 | -1 |
| 0 | 0 | 0 |
| 0 | 1 | 3 |
| 0 | 2 | 5 |
| 1 | -1 | 0 |
| 1 | 0 | 1 |
| 1 | 1 | 5 |
| 1 | 2 | 9 |
| 2 | -1 | 3 |
| 2 | 0 | 5 |
| 2 | 1 | 9 |
| 2 | 2 | 15 |
So, the table for the operation on the set {-1, 0, 1, 2} is as given above.