Question 1205232: for the relations: A x B, such that:
A = {-7,-6,..., -4} and B = {42,43,...,45}
count how many possible ways N1 to have functions p = A --> B
N1 =
count how many possible ways N2 to have p as one to one function.
N2=
count how many possible ways N3 to have p NOT onto.
N3 =
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
A = {-7, -6, -5, -4}
B = {42, 43, 44, 45}
Each set has 4 elements.
It's not clear why your teacher used the triple dot notation when it's not needed.
For the input x = -7, from set A, there are 4 choices it could map to in set B. Same with x = -6 and so on.
There are 4*4*4*4 = 4^4 = 256 different functions possible.
A function is a collection of (x,y) points where x doesn't repeat itself.
The y values can repeat, but the function wouldn't be one-to-one.
Often the (x,y) points are connected with some kind of line or curve to form a graph; however, in this case, the points will be separate disconnected islands.
A function is considered one-to-one (aka injective) when we don't have repeated outputs. This is when each input is assigned a unique output.
You can think of it like pairing up dance partners.
The input x = -7 has 4 choices to pick from.
After we pick something from B, there are 4-1 = 3 choices left for the next input. Then 3-1 = 2 choices for the third input, and finally 1 choice for the last input.
There are 4*3*2*1 = 24 one-to-one functions possible.
n(A) = number of values in set A
Because n(A) = n(B), a function that is onto is automatically one-to-one, and vice versa.
The proof is left as an exercise for the student.
"Onto" is the same as "surjective". They both refer to the idea that every item in set B is targeted.
Here's an example of a surjective function
{(-7,42),(-6,43),(-5,44),(-4,45)}
and here's something that isn't surjective.
{(-7,42),(-6,42),(-5,44),(-4,45)}
in the 2nd example, the output y = 43 isn't targeted in set B.
Therefore, the 2nd example isn't surjective. Notice how the 2nd example isn't injective either since we have a repeated y value.
The term bijective means "both injective and surjective".
Here's an article that talks about the terms mentioned
https://www.mathsisfun.com/sets/injective-surjective-bijective.html
We found there are 24 bijective functions and 256 functions possible.
That must mean there are 256-24 = 232 functions that are not surjective.
Answers:
N1 = 256
N2 = 24
N3 = 232
|
|
|
