Boolean expression/graphs/strings

adityanal

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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.
 
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: \(\displaystyle \neg[(P\to Q)\wedge(Q\to P)\}\)
 
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.
You should know by now that to get help you must show us your work so we know where you are stuck.
 
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