For each of the following functions, determine whether the function is: Injective (one-to-one). Surjective (onto). ⚫ Bijective. Justify your answers. 1.3 f: R→ ZZ such that f(x) = [4x] (i.e., ceiling of 4x).

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}

Determine whether each of the following functions f : {a,b,c,d} -> {a,b,c,d} is one-to-one and/or onto.  (a) f(a) = b, f(b) = a, f(c) = b, f(d) = c (b)  f(a) = b, f(b) = b, f(c) = d, f(d) = c (c)  f(a) = b, f(b) = a, f(c) = c, f(d) = d (d)  f(a) = d, f(b) = a, f(c) = c, f(d) = b (e)  f(a) = c, f(b) = d, f(c) = a

7. Let P denote the set of all phones in the world such that p ∈ P is a phone. Thus, S(p)denotes that “p is a SamsungTM phone”, N(p) denote that “p is a NokiaTM phone”, and G(p)denote that “p is a GoogleTM phone”. Therefore, express each of the following statements usingquantifiers, logical operations, and the propositional functions: S(p),N(p),G(p).(a) There is a GoogleTM phone that is also a SamsungTM phone.(b) Every NokiaTM phone is a GoogleTM phone.(c) No NokiaTM phone is a GoogleTM phone.(d) Some NokiaTM phones are also SamsungTM phones.(e) Some NokiaTM phones are also GoogleTM phones and some are not. PS: Please do not answer them in a paper format.