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Which field of Discrete Mathematics ?
Number Theory,Combinatorics,Set Theory ,Probablity,Operations Research,Game theory, decision theory, utility theory, social choice theory,Discretization,Hybrid discrete and continuous mathematics...........................There are many which one?
I am currently taking a Discrete Mathematics for Computing course, and was looking for some help with different problems from that course. We are currently working on propositional logic and I was having a little trouble with a couple of problems.
There is a discrete mathematics group, but there is practically no activity in it, so I thought I'd give this group a try to see if someone was available that might could help out. Here's one of the problems that I can't figure out. "Express these statements using logical operators, predicates, and quantifiers. a) Some propositions are tautologies. b)The negation of a contradiction is a tautology. c)The disjunction of two contingencies can be a tautology. d)The conjunction of two tautologies is a tautology. The answer supposed that T(x) is a tautology and C(x) is a contradiction. The answer for A is\[\exists x T(x)\] just so you get the form they are looking for in the answers. I got A without any problem, but I didn't get the answers provided in the back of the book for the other three problems. b) I got \[\forall x ( \not C(x) \rightarrow T(x))\]the book said \[\forall x ( C(x) \rightarrow T( \not x)\]. I couldn't find the regular negation sign in the equation editor so the slash through the letter indicates negation. What does my answer not work? For the other two, I did not get the given answers at all. What would you get for these? Thanks for your help!
You should try the Mathematics group.
a) Some propositions are tautologies. Tautologies being defined as a statement form that is always true regardless of truth values of the individual statements substituted for its statement variables. Say you have this proposition: p v ~p. Whatever boolean type you give "p" this will always logically be true. b)The negation of a contradiction is a tautology. This goes back to what a negation is, which is just the opposite of a given proposition. A contradiction is a proposition that will always come out false, such as "p and ~p" If a contradiction is always false, the negation would have to be a statement that is always true, which is a tautology. c)The disjunction of two contingencies can be a tautology. Such as I stated in question A. If you have p OR not p. You truth values will always be true, or a tautology. d)The conjunction of two tautologies is a tautology. If you conjoin two tautologies with an "AND" operator. The truth value will always be true, which is obvious. True AND True is always True. Since a tautology is always true. ∀x(C(x)→T(/x), Think about what the book is saying. "For all values 'x', The contradiction of x implies the negation of a tautology. I'm not too sure if I truly (pun intended) answered any of your questions, but I hope I at least helped!