Review 232 midterm forgotten terms

Consistent

1. P ∨ Q 2. ¬P 3. P→Q when P = F and Q = T all three statements are T so the argument (the specification is consistent)

What is a partial function ?

A partial function is one where not all elements of the domain are mapped to the codomain

Difference between sound and valid ?

A valid argument means the premises necessarily lead to the conclusion. For instance, "1 = 2, 3 = 1, therefore 2 = 3." Notice that this has nothing to do with the truth of the premises only that the conclusion must be true based on the premises. A sound argument is both valid and the premises are true.

what is a total function ?

All elements of the domain are mapped to at least one element in the codomain.

What is a one-to-one function ?

Also known as injection is a function that maps distinct elements of its domain to distinct elements of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain. A partial function can be injective, not all elements of the domain or co-domain need to be represented but all preimages that are mapped to the codomain have at most one image in the codomain.

a consistent argument

Are two arguments that can both be true in the same universe? in this example P && Q and P or Q are both true in a world where both p and q are true. Consequently, there exists a world where their truth values are the same and they are both T so P&&Q and P or Q are consistent. If looking at multiple propositions, can you assign truth values to the variables in the propositions such that all of the propositions are true. If you can do this, then the propositions and the argument is consistent.

What is a bijection ?

In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

What are inconsistent propositions ?

Inconsistent propositions are two propositions that are never true at the same time I.e. when the same truth values are assigned to each proposition, the result is never the same they can both be false but never both TRUE

when does an inverse function exist?

Only if the function is a bijection

sound argument

a valid argument in which all of the premises are true NOTE this is different than a consistent argument, where it does not matter if the statements/premises in the argument are T or F as long as the final statement/i.e. conclusion is T An argument consists of a sequence of statements called premises and a statement called a conclusion

what is an onto function ?

also known as surjective means that every value in the codomain has at least one element in the domain such that F(x) = y

What is a valid argument ?

logical argument is valid if its premises logically imply its conclusion; that is, the argument is valid if the conclusion must be true on the assumption that the premises are true. (i.e. it's a tautology ? ) No matter what the truth values in the premises T or F the conclusion is always true So it doesnt matter if one of the statements example (P and Not P) is false, if all of the statements lead to a T conclusion than it is a valid argument. NOTE: this is different than a sound argument where all of the premises Must be true AND the conclusion must be true