Logic 1Recall that in Engr 121 we used arithmetic operations instead of , and
- What was the symbol for each? should be natural, the others a little forced
- Draw a truth table for and determine which arithmetic operation is appropriate An inequality
Logic 2On first glance, the statements and appear to say different things.
- Compute a truth table for both. What is the result?
- Similarly, for each statement, find all situations where it is false. What do you see here?
- Rewrite both statements using the law of contrapositive. Is this clearer?
- Directly prove that if is true, then is true. You might need cases Assume either or is true .....
- It is difficult to prove the converse statement, if , then . Give it a go! Focus on , consider the possibilities Either is true, or it is false. One case is easy ..
- Explain why allows to be true.
- What does say?
- What is the negation of the previous statement?
- This is a statement of what induction does, the set is the collection of natural numbers collected by the process This is very complicated!
- Which part is the base case?
- Which part is the induction hypothesis and step?
- The last part is a promise, what is the promise?
- What is the negation of the statement and what does it say? There is some subset of for which induction doesn't work, it misses some number.
- The least number principle (which we haven't seen) says that every non-empty subset of has a least element
- Which part talks about the set being non-empty?
- What is the least element?
- Negate this statement and explain what is says There is a subset of , no matter where we think the bottom of the set is, we can always go lower.
- Use induction to prove that the least number principle is true. Look at , the set of values that doesn't collectDo induction on , you should find that it satisfies the conditions, so it collects everything ...