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Show that there is a language a ⚆ {0, 1} â— with the following properties: 1. for all x â a, |x| ≤ 5. 2. no dfa with fewer than 9 states recognizes a. hint: you don’t have to define a explicitly; just show that it has to exist. count the number of languages satisfying (1) and the number of dfas satisfying (2), and argue that there aren’t enough dfas to recognize all those languages. to count the number of languages satisfying (1), think about writing down all the strings of length at most 5, and then to define such a language, you have to make a binary decision for each string about whether to include it in the language or not. how many ways are there to make these choices? to count the number of dfas satisfying (2), consider that a dfa behaves identically even if you rename all the states, so you can assume without loss of generality that any dfa with k states has the state set {q1, q2, . . , qk}. now think about how to count how many ways there are to choose the other four parts of the dfa.

Usually, when we sniff packets, we are only interested certain types of packets. we can do that by setting filters in sniffing. scapy’s filter use the bpf (berkeley packet filter) syntax; you can find the bpf manual from the internet. set the following filters and demonstrate your sniffer program again (each filter should be set separately): (a) capture only the icmp packet. (b) capture any tcp packet that comes from a particular ip and with a destination port number 23. (c) capture packets comes from or to go to a particular subnet. you can pick any subnet, such as 128.230.0.0/16; you should not pick the subnet that your vm is attached to.

Budgets you to do all of the following expect a) send frivolously b) avoid over spending c) gain financial independence d) examine your priorities and goals