Help Which of the following sets of ordered pairs represent functions? Which are

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Which of the following sets of ordered pairs represent functions? Which are onto? Which are one-to-one? Finally, describe each relation's inverse. (Recall that Z^+ is the set of positive integers.) {(1, 1), (2, 4), (3, 3), (3, 4), (4, 5), (5, 5)} {1, 2, 3, 4, 5} times {1, 2, 3, 4, 5} ii) {(1, 4), (2, 4), (3, 4), (4, 4), (5, 4)} {1, 2, 3, 4, 5} times {1, 2, 3, 4, 5} iii. {(x, y) Z^+ times Z^+ |gcd(x, y) = 2} iv) {(x, y) N times (-1, 0, 1}|y = sin(x d/1)} Given the following weighted graph, fine the minimum spanning tree via each method. (There will only need to be lour edges added. They will even be the same four. The algorithms just pick them in a different order.)

Explanation / Answer

i) Not a function because looking at (3,3) and (3,4),
there's a value of x at which there are two values of y
So, not a function

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ii)
Yes, a function because there are no 2 y-values for the same x-value
Not one-to-one because for the same y-value, we have multiple x-values
Not onto either because the entire range [1,2,3,4,5] is not being used up

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iv)
y = sin(x*pi/2) will only take values between [-1 , 1]
Now, x is natural. So, sin(x*pi/2) can only be -1, 0 or 1
Yes, this is a function
Not one to one because for the same y-value, we'd have multiple
x-values surely as sine function is periodic, as in repeats itself
Definitely onto because the entire range {-1,0,1} is used u

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