A sentence that has a definite state of being true or false

Suppose is a statement. Then the negation of , denoted by , is the statement asserting the opposite truth value to .

Ie, the double negation of is logically equivalent to

A sentence that contains a variable, where the truth of the sentence is determined by the value of the variable chosen from the domain of the variable

1. A universally quantified statement is of the form

- Read as "For all in , is true"
2. An existentially quantified statement is of the form

- Read as "There exists (at least one) value of in for which is true"

Where is false, is true.

where is true (), is false.

Quantifiers will cover the entire scope they are in

Let be a statement. Note that either or must be false, so the compound statement is always false.

• " is true" is a contradiction
• is always false

An assertion that is true in every interpretation

• is always true
• is always true

To prove chose a representative mathematical object . This cannot be a specific object. It has to be a placeholder. Then, show that P must be true for , using known facts about

case analysis can be used for different parts of the domain

To prove , start from and show the steps to arrive to the desired result.

Constructive proofs provide a method for creating the desired object. Usually, direct proofs are constructive proofs.

To disprove , find an element for which is false.

To prove the , provide an explicit value of from the domain that satisfies

1. Existence: prove that there is at least one element such that is true (i.e prove )
2. Uniqueness: do one of the following
   1. Assume and are true for , and prove that this assumption leads to the conclusion
   2. Assume and are true for distinct (so ), and prove that this assumption leads to a contradiction

To prove the existentially quantified statement , prove the universally quantified statement