Describe a rule of inference that can be used to prove that there are exactly two elements x and y in a domain such that P(x) and P(y) are true. Express this rule of inference as a statement in English.
Answers
Rules of Inference and Formal Proofs
Proofs in mathematics are valid arguments that establish the truth of
mathematical statements.
An argument is a sequence of statements that end with a conclusion.
The argument is valid if the conclusion (final statement) follows from
the truth of the preceding statements (premises).
Rules of inference are templates for building valid arguments.
We will study rules of inferences for compound propositions, for quantified
statements, and then see how to combine them.
These will be the main ingredients needed in formal proof
Valid Arguments using Propositional Logic
Consider the following argument (sequence of propositions):
If the prof offers chocolate for an answer, you answer the prof’s
question.
The prof offers chocolate for an answer.
Therefore, you answer the prof’s question.
Let p be “the prof offers chocolate for an answer”
and q be “you answer the prof’s question”.
The form of the above argument is:
p → q
p
∴ q
The argument is valid since ((p → q) ∧ p) → q is a tautology
Proof methods and Informal Proofs
After studying how to write formal proofs using rules of inference for
predicate logic and quantified statements, we will move to informal
proofs.
Proving useful theorems using formal proofs would result in long and
tedious proofs, where every single logical step must be provided.
Proofs used for human consumption (rather than for automated derivations
by the computer) are usually informal proofs, where steps are combined
or skipped, axioms or rules of inference are not explicitly provided.
The second part of these slides will cover methods for writing informal
proofs.
Answer:
Dear Grandpa and Grandma,
How are you ? I hope you are in the pink of your health. I am fine here. It's been a long time since we met. If it would be possible for you to visit the apple of your eye,come here. I miss you. However, i am writing this letter to you describing about my summer vacations.
This vacation was a vanity of memorable moments. As you know, it's my favorite activity to spend time at the orphanage and so did I. There actually I learned how miserable situations can come across and how you have to taste the bitter truth. But astonishingly, they all seemed happier than me. I have so many complaints but they are all very satisfied. Not only the orphanage was the part of my memories. I also visited the "Galaxy Museum" , just as grandpa had advised me . Really ! Its a megestic place. I also went at beaches with my friends for swimming. It was a great fun.This time as I wanted to do something different from the past vacations, I had participated in science astronomy competition which was held by our area's foundation group. And guess what ??? I won second prize for my project. It was a prideful moment for me. But alas ! nobody knows how time passes off. soon my vacation will get over
Last but not the least, my regards to both of you. I will be waiting for you reply too !
Your lovely grand daughter,
XYZ.