Question 1. Consider the statement 'For all integers a,b and c, if ab|c| then a|c| and b|c|.' Determine whether the statement is true or false. If it is true, prove the statement directly from the definitions. If it is false, give a counterexample.
Question 2. Consider the statement 'For all integers a,b and c, if a|bc| then a|b| or a|c|.' Determine whether the statement is true or false. If it is true, prove the statement directly from the definitions. If it is false, give a counterexample.
Question 3. Let R be the statement: The square root of any irrational number is irrational.
a) Write a negation for R.
b) Prove R by contradiction.
Question 4. Let's return to the island of knights and knaves. Recall that each inhabitant is either a knight or a knave (but not both). Knights always make true statements, while Knaves always make false statements. You meet three inhabitants: Carol, Dan, and Eve and ask some questions.
You ask Carol: "How many knights are among you three?" Carol answers, but you do not hear her.
Then you ask Dan: "What did Carol say?" Dan replies, "Carol said there is one knight among us."
Then Eve interjects "Dan is lying!"
What are the identities of Dan and Eve? Explain.
What are the identities of Dan and Eve? Explain. Then Eve interjects "Dan is lying!" Us." hear her b) Prove R by contradiction. a) Write a negation for R. Determine whether the statement is true or false. the definitions. If it is false, give a counterexample. make false statements. You meet three inhabitants: Carol, Dan, and Eve and ask some questions. a knight or a knave (but not both). Knights always make true statements, while Knaves always Question 4. Let's return to the island of knights and knaves. Recall that each inhabitant is either Question 1. Consider the statement For all integers a, b and c, if ab ( c then a | c and b | c.