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Logic

Page history last edited by Andrea Grieser 10 years, 6 months ago

In mathematics, we cannot simply use unproven suppositions or assertions in calculations or discussions.  We must use logical reasoning in all that we do.

 

We propose conjectures, statements based on observation, which we then must prove.  We prove conjectures using axioms, postulates, and theorems.  Once a conjecture is proven, it becomes a theorem, which can then be used to prove other conjectures. 

 

Two types of reasoning we can use to prove conjectures is inductive reasoning and deductive reasoning.

 

Proofs can be presented in different ways, including two-column proofs and paragraph proofs.

 

Conjectures, axioms, postulates and theorems are often the form of conditional statements.  A conditional statement is a logical statements that has a hypothesis and conclusion.  A common type of conditional statement is an "IF-THEN" statement.  The hypothesis is the phrase after the "if" and the conclusion is the statement after the "then."  See the example below...

 

 

Conditional statements have a truth value of either true or false.  If the hypothesis being true makes the conclusion true, then the truth value of the statement is true.  Otherwise, if a true hypothesis produces a false conclusion, then the statement is false.  We can often prove a conditional statement is false by finding a counterexample, that is, an example that shows the conclusion is false.  For example, the statement "if an animal has four legs then it is a dog" can be proven false because we can find a counterexample, such as cat or horse, which has four legs but is not a dog. 

 

Conditional statements with the same truth value are said to be equivalent statements.

 

The converse of a conditional statement occurs when we swap the hypothesis and conclusion.  The inverse of a conditional statement occurs when we negate (take the opposite of) both the original hypothesis and conclusion.  The contrapositive of a statement occurs when we take the inverse and converse of a statement. 

 

Given  the statement "If an animal meows then it is a cat", the...

  • converse is:  If an animal is a cat then it meows
  • inverse is:  If an animal does not meow then it is not a cat
  • contrapositive is:  If an animal is not a cat then it does not meow.

 

A conditional statement is biconditional if both the original statement and its converse are true.  In the example above, the statement is biconditional because both the original statement (if an animal meows then it is a cat) and its converse (if an animal is a cat then it meows) are both true, assuming by "cat" we mean domestic cat (because lions don't meow, which would make the original statement false!).

 

Biconditional statements use the words "if and only if," abbreviated as IFF.  An example is shown below...

 

 

The original statement and its contrapositve are always equivalent (that is, they have the same truth value).

 

Notation

We use notation conventions in describing logical statements.

 

A conditional statement with hypothesis p and conclusion q may be written as p -> q, read "p implies q"

To negate a statement, hypothesis, or conclusion, we use the "~" symbol.  For example, ~p -> ~q may be read as "not p implies not q" and is the inverse of statement p -> q.   For biconditional statements, we use a double-sided arrow, as in p <-> q, read as p IFF q.

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