49 lines
1.7 KiB
Markdown
49 lines
1.7 KiB
Markdown
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Strings and languages:
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- An alphabet is a finite set of symbols, such as {1,2,3} or {0,1}
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- A string is an alphabet $\sum$ is a finite sequence of symblols from $\sum$, for example 010010 and 10010101 are strings over the alphabet {0,1}
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- A languge is a set of strings
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- L1 = {00, 01, 10, 11}
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- L2 = {a^n | n is an integer non-negative}
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Notation: a to the power of n just means a repeated n times
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Examples
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$L_8 = \{a^m b^n | m,n \in \mathbb{z}^{nonneg}\}$
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So valid strings would be
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aaabbb, aaabbbb, ab, abb
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$L_9 = \{a^n b^n | n \in \mathbb{z}^{nonneg}\}$
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So valid strings would be
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aabb, ab, aaaabbbb, aaaaaabbbbbb
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$L_{11} = \{w \in \{a, b\}^* | \text{The number of "a"s is equal to the number of "b"s}\}$
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This would include everything that L9 includes, but allows for the alphabet can be out of order
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You can also written this as
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$L_{11} = \{w \in \{a, b\}^* | n_a(w) = n_b(w)\}$
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You can extend the repetition notation (exponent) to include multiple symbols by wrapping it in parentheses
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Example:
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$L_{12} = \{(ab)^n | n \in \mathbb{z}^{nonneg}\}$
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This means we need to be careful as
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$L_{13} = \{ab^n | n \in \mathbb{z}^{nonneg}\}$
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is different, as L12 is asking for repetition of ab, while L13 is assigning a prefix of a and repetition of b
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$L_{14} = \{a^i b^j c^k | i,j,k \in \mathbb{z}^{nonneg}, j = 2i + 3k\}$
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So this means that for any number of "a"s and "c"s, the number of "b"s is predetermined if you have selected your "a"s and "c"s
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Intro to functions and relations
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Informally, a function (f) descrives an input output situation (A -> B)
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For every input element a in A, there is exactly one ouput element b in B
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f maps a to b
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A is the domain of f
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B is the co-domain of f
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