Boolean expression/graphs/strings

[adityanal]

1.Boolean Logic: Give a Boolean expression consisting of only P’s, Q’s, ¬’s, ∧’s, and ∨’swhich is logically equivalent to the Boolean expression below:

¬ (P ↔ Q)​

2.Graphs: Is the statement “For every natural number n ≥ 1 there exists a directed graphof n vertices for which every vertex has an indegree equal to its outdegree” TRUE or FALSE?


3.Strings and Languages: For alphabet Σ = {a, b, c}, suppose x ∈ Σ∗ and |x| = 5. Give astring x0that is a substring of x and has the following property:Among all substrings of x, x0is both a prefix of x and a suffix of x, and is the longest substringof x.

 

[pka]

1.Boolean Logic: Give a Boolean expression consisting of only P’s, Q’s, ¬’s, ∧’s, and ∨’swhich is logically equivalent to the Boolean expression below: ¬ (P ↔ Q)

Can the new expression contain parentheses? For EXAMPLE: \(\displaystyle \neg[(P\to Q)\wedge(Q\to P)\}\)

 

[Steven G]

1.Boolean Logic: Give a Boolean expression consisting of only P’s, Q’s, ¬’s, ∧’s, and ∨’swhich is logically equivalent to the Boolean expression below:

¬ (P ↔ Q)​

2.Graphs: Is the statement “For every natural number n ≥ 1 there exists a directed graphof n vertices for which every vertex has an indegree equal to its outdegree” TRUE or FALSE?


3.Strings and Languages: For alphabet Σ = {a, b, c}, suppose x ∈ Σ∗ and |x| = 5. Give astring x0that is a substring of x and has the following property:Among all substrings of x, x0is both a prefix of x and a suffix of x, and is the longest substringof x.

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