Determine if each function is injective, surjective or bijective. Give one counterexample for each that it is not. 1. function f from {a,b,c,d} to itself, where f(a) = d, f(b) = b, f(c) = a, f(d) = c
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Determine if each function is injective, surjective or bijective. Give one counterexample for each that it is not.
1. function f from {a,b,c,d} to itself, where f(a) = d, f(b) = b, f(c) = a, f(d) = c
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- Let A = {1, 2,3} and B = {a, b, c, d} What is the function from a to b?Which functions are one-to-one? Which functions are onto? Describe the inversefunction for any bijective function.(a) f : Z → N where f is defined by f (x) = x4 + 1(b) f : N → N where f is defined by f (x) = { x/2 if x is even, x + 1 if x is odd}(c) f : N → N where f is defined by f (x) = { x + 1 if x is even, x − 1 if x is odd}6. X is (a,b,c,d), Y is {1,2,3,4). {(a,1), (b,2), (c,2), (d,4)) is a function from X to Y.
- Rewrite the following function definitions using lambda notation: f(x) = x + 1 f(x) = xFor all a and b in the domain of a function f, the function is injective iff f(a) #f(b) a=Db f(a)=f(b) a=b f(a)=f(b) atb o fly)=x iff f(x)=y A Moving to another question will save this response. 5Consider the following functions. Decide whether these functions are injective, surjective, and invertible. Justify your answer (e.g., if you claim that a function is invertible, you need to give a justification as to why you think that function is invertible). Give counterexamples when needed.
- write all functions f: {1,2} -> {a,b,c,d} (in two line notation) how many functions are there how many are surjective how many are injective how many are bijective please helpQl: The Collatz conjecture function is defined for a positive integer m as follows. (COO1) g(m) = 3m+1 if m is odd = m/2 if m is even =1 if m=1 The repeated application of the Collatz conjecture function, as follows: g(n), g(g(n)), g(g(g(n))), ... e.g. If m=17, the sequence is 1. g(17) = 52 2. g(52) = 26 3. g(26) = 13 4. g(13) = 40 5. g(40) = 20 6. g(20) = 10 7. g(10) = 5 8. g(5) = 16 9. g(16) = 8 10. g(8) = 4 11. g(4) = 2 12. g(2) = 1 Thus if m=17, apply the function 12 times in order to reach m=1. Use Recursive Function.Define a function make_derivative that returns a function: the derivative of a function f. Assuming that f is a single- variable mathematical function, its derivative will also be a single-variable function. When called with a number a, derivative will estimate the slope of f at point (a, f(a)). the Recall that the formula for finding the derivative of f at point a is: f'(a) = lim h→0 where h approaches 0. We will approximate the derivative by choosing a very small value for h. The closer h is to 0, the better the estimate of the derivative will be. def make_derivative (f): """Returns a function that approximates the derivative of f. >>> def square (x): f(a+h)-f(a) h Recall that f'(a) = (f(a + h) - f(a)) / h as h approaches 0. We will approximate the derivative by choosing a very small value for h. # equivalent to: square = lambda x: x*X return x*x >>> derivative = make_derivative (square) >>> result = derivative (3) >>> round (result, 3) # approximately 2*3 6.0 |||||| h=0.00001 "***…
- Q3: Two Variables es are define by the following equations: f1=E m(2,4,5,10, 12,13,14). f2=Em (2,9,10,11,13,14,15). Show how these functions can be implemented on a PLADefine Functions find and find_if.3. (a) Consider the following algorithm. Input: Integers n and a such that n 20 and a > 1. (1) If 0SEE MORE QUESTIONS