Math 13 Ch3
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Logic is the study of what's true or false or somewhere in between. It is important as we study logic that we will be most concerned with the truth of falsity of the argument rather than the content of the argument.
- A Statement in logic is a declarative sentence that is either true or false. We represent declarative sentences by lowercase letters such as p, q, or r.
- Example: p: Mike is 6 years old.
- A simple statement contains a single idea.
- Example: Mike is 6 years old.
- A compound statement contains several ideas combined together. The words used to join the ideas of a compound sentence are called connectives.
- Example: Mike is 6 years old and acts like an older child.
- There are many connectives in the English language. In logic, connectives fall into five categories.
- Negation (not or the removal of not)
- Conjunction (and)
- Disjunction (or)
- Conditional (if . . . then)
- Biconditional (iff; if and only if)
- Negation is a statement expressing the idea that something is not true. We represent negation by the symbol ~. Remember Compound statements have connectives and negations are considered a connective. Also that negation means the oppisit if what was said; meaning you can either add or remove "not".
- p: Santa is real. Santa is not real.
- q: Robert Frost is not a great poet.
- Robert Frost is a great poet.
- A conjunction expresses the idea of and. We use the symbol ^ to represent the conjunction.
Are ~(p^q) and ~p ^ ~q equvilant?
- A disjunction expresses the idea of or. We use the symbol v to represent a conjunction.
- A conditional expresses the notation of if...then. We use the arrow, --> to represent a conditional.
- A biconditional represents the idea of if and only if or iff. Its symbol is the double arrow <-->.
- In addition to connectives, there are special words called Quantifiers that you need to understand a sentence. Quantifiers tell us "how many" and fall into two categories.
- Universal Quantifiers are words such as all and every that state that all objects of a certain type satisfy a given property.
Existential quantifiers are words such as some, there exists, and there is at least one that state that there are one or more objects that satisfy a given property.
- Negating universal quantifiers is specific and different than nagating existential quantifiers. You must have a good understanding of what each means.
Other way of doing a truth table process.
- Each number represents the outcome using that connective, simplifying it.
If the final column of a truth table contains all T's, then the statement is always true (valid). Such a statement is called a tautology.
- If p and q are statements, then
- ~(p^q) is logically equivalent to ~p v ~q
- ~(p v q) is logically equivalent to ~p ^ ~q
Two statements are logically equivalent if they have the same variables and when their tables are computed, the final columns in the tables are identicle.
An argument is a series of statements called premises followed by a single statement called the conclusion. If the argument is valid then the truth table will be a tautology. If there are any falses in the final column, then the argument is invalid.
Premises to table
- transitioning from each row we use and, ^ and for the therefore we use if..then arrow. Note: tautology.
What are the exceptions or rules for:
- ~ is oppisite
- ^ is TTT
- v is FFF
- -> (if..then) is TFF
- <--> (iff) is pairs are T
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