Define the function f: {0, 1}3→{0, 1}3. The output of f is o…

Define the function f: {0, 1}3→{0, 1}3. The output of f is obtained by taking the input string and replacing the first bit by 1, regardless of whether the first bit is a 0 or 1. For example, f(001) = 101 and f(110) = 110.  Under this function, what is the preimage of 111?

Suppose you have two unknown functions r{“version”:”1.1″,”ma…

Suppose you have two unknown functions r{“version”:”1.1″,”math”:”\(r\)”} and s{“version”:”1.1″,”math”:”\(s\)”}, both defined on R+{“version”:”1.1″,”math”:”\(\mathbb{R^+}\)”}.  If r=Θ(n){“version”:”1.1″,”math”:”\(r=\Theta(n)\)”} and s=Θ(n4){“version”:”1.1″,”math”:”\(s=\Theta(n^4)\)”}, what can you say about the asymptotic growth rate of  s ⋅ r 2 {“version”:”1.1″,”math”:”\(s \cdot r^2\)”}?

On the Bahamian island of Andros, mosquitofish populations l…

On the Bahamian island of Andros, mosquitofish populations live in various, now-isolated freshwater ponds that were once united. Currently, some predator-rich ponds have mosquitofish that can swim in short, fast bursts; other predator-poor ponds have mosquitofish that can swim continuously for a long time. When placed together in the same body of water, the two kinds of female mosquitofish prefer to breed only with males that have their same swimming ability. Which of the following phenomena have helped promote divergence between the mosquito populations?