# CMI2020-A: 3

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Some children are given boxes containing sweets. Harish is happy if he gets either gems or toffees. Rekha is happy if she gets both bubble gums and peppermints. Some of the boxes are special, which means that if the box contains either gems or toffees, then it also contains bubble gums and peppermints. If Harish and Rekha are given boxes that are not special, which of the following can we infer?

1. Harish is happy
2. No bubble gums in Rekha’s box
3. No toffees in Harish’s box
4. There are peppermints in Rekha’s box
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Which of the following languages over the alphabet $\{0,1\}$ are $not$ recognized by deterministic finite state automata $(DFA)$ with $three$ states? Words which do not have $11$ as a contiguous subword Binary representations of multiples of three Words that have $11$ as a suffix Words that do not contain $101$ as a contiguous subword
Consider the following regular expressions over alphabet$\{a,b\}$, where the notation $(a+b)^+$ means $(a+b)(a+b)^*$: $r_1=(a+b)^+a(a+b)^*$ $r_2=(a+b)^*b(a+b)^+$ Let $L_1$ and $L_2$ be the languages defined by $r_1$ and $r_2$, respectively. Which of the following regular expressions define $L_1\cap L_2$ ... $(a+b)^*a\;b(a+b)^*$ $(a+b)^*b(a+b)^*a(a+b)^*$ $(a+b)^*a(a+b)^*b(a+b)^*$
A boolean function on $n$ variables is a function $f$ that takes an n-tuple of boolean values $x \in \{0,1\}^n$ as input and produces a boolean value $f(x)\in \{0,1\}$ as output. We say that a boolean function $f$ ... of symmetric boolean functions on $n$ variables? $n+1$ $n!$ $\displaystyle \sum^n_{i=0} \begin{pmatrix} n\\i \end{pmatrix}$ $2^{n+1}$