Chapter 5 - Existence and Proof by Contradiction

Disproving an implication:

You must show that there exists some element x that is part of the domain and satisfies the condition, but leads to a false conclusion

Uniqueness Theorem

a particular type of existence problems - there exists EXACTLY ONE object having certain properties

Note

an open sentence R(x) that is false over some domain S may very well be true over a subset of S. Therefore, the truth (or falseness) of a statement for all x that is an element of S, R(x) depends not only on the open sentence R(x) but on its domain as well

Disproving Uniqueness

Basically find more than one example that satisfies the desired property

Uniqueness Proof Approach

First, claim the existence of one object that satisfies the desired property - Now, consider two arbitrary objects satisfying the property and prove that these two objects turn out to always be = and so the one object is unique

Universal Quantifier

For all x which is an element of S, R(x)

Existence Theorem:

In an existence theorem the existence of at least one object possessing some specified property or properties is asserted - basically, this means that there exists some x that is an element of S such that R(x) - You can just think of an example and show that it "satisfies the desired property"

Counterexample/Disprove Proof

Provide a counterexample and show that the element leads to the conclusion being false

Proof by Contradiction:

Show that the premises under the given domain leads to a conclusion that is a contradiction, whether that contradiction is relevant to the given statements in the proof or some other mathematical definition/axiom - begins by assuming the existence of a counterexample of this quantified statement - "Assume, to the contrary, that..." the premise stays the same, but now the conclusion is false - end with "This, however, is a contradiction."

Counterexample

The element x of the set S for which R(x) is false

Note

The opposite of < is >=

Existential Quantifier

There exists some x which is an element of S such that R(x)