$1.$ Add the following two$$s complement binary numbers of $8$ bits and also express the answer in decimal. Identify each case in which the answer is incorrect because of overflow.

a$.00001111+10011001$

b$.11111000+11101111$

$2.$ Consider the following addition problems. Please first convert each decimal number into eight$-$bit twos complement notation, perform the addition in twos complement notation, then convert the results back to decimal numbers. Identify each case in which the answer is incorrect because of overflow.

a$.32+54$

b$.-36-45($hint: you can interpret this as $(-36)+(-45)$

c$.-123-45$

d$.125-124$

$3.$ Let X$,$ Y$,$ and Z$,$ are three sets, prove that X$($Y$$Z$)=($X$$Y$)($X$$Z$).$

$4.$ Design a DFA that accepts strings starting with either $\text{'}10\text{'}$ or $\text{'}01\text{'}$

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$5.$ Design a DFA that recognizes strings containing the substring $\text{'}010\text{'}.$ For instance, strings like $\text{'}010111\text{'},\text{'}11011001101000111\text{'},$ etc., should be accepted.

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$6.$Construct a Deterministic Finite Automaton $($DFA$)$ that recognizes strings with an odd number of symbols. For instance, $\text{'}001\text{'},\text{'}11001\text{'},$ and $\text{'}1000110\text{'}$ are examples of strings accepted by the DFA.

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