A proposition is a statement of either true or false. The purpose is to analyze these statements either individually or in a composite manner. A propositional consists of propositional variables and connectives. If a proposition is true we can say its true value is true, and if a proposition is false, we can say its true value is false.
Many mathematical statements are constructed by combining one or more propositions, new proposition called compound propositions,are formed from existing propositions using logical operations .
6 + 9 = 18, is a proposition i.e. false.
Pound Sterling is the currency of United Kingdom, is a proposition of true statement.
Following is not a proposition
“A is less than 2 “ because unless we give a specific value of A and cannot say whether the statement is true or false.
Truth function is the truth value that can be observed in truth table with different propositional logic statements. A function that determines the truth value of a complex sentencesin the term of truth values of the component sentences without reference to their meaning.
Five connectives are - OR (∨), AND (\(\wedge\)), Negation/NOT (\(\lnot\)), Implification/If then (\(\to\)), If and only if (\(\Leftrightarrow\)).
|A||B||A \(\vee\) B|
|A||B||A \(\wedge\) B|
The rule of logic give meaning to mathematical statements are called proposition logic.
These rules are used to distinguish between valid and invalid mathematical arguments.
Proposition logic is also know as sentential logic or statemental logic that studies ways of joining and/or modifying entire propositons to form more comfortable propositions. In propositional logic the simplest statements are considered as indivisible units, and hence, the propositional logic doesnot study those logical properties and the relations that depend upon parts of statements.
It largely involves studying logical connectives as the words “AND “ and “OR” and the rules determining the truth values of the propositions they are used to join. These rules mean for validity of arguments and such logical relationships between statements as being consistent or in consistent with one another as well as logical properties of of propositons.
Propositon can be thought as a logical operators.A logical operator is any word used either to modify one statement to make a different statement or join together to form more complicated statement.Such as AND ,OR,NOT,If then,If and only if, are all operators.Some logical operators are not truth functional.
For example, both of the operations are true:
2 + 2 = 4.
Someone is reading an article in a philosophy encyclopedia.
Consider the corresponding statements modified with the operator “necessarily”:
Necessarily, 2 + 2 = 4.
Necessarily, someone is reading and article in a philosophy encyclopedia.
Here, the first example is true and the second is false.True or false of a statement using the operator “necessarily” does not depend entirely on the true or false of the statement modified.
Expressing statements in Logic Propositional Logic
Let p be the proposition. The negation of p, denoted by \(\lnot\)p.
Suppose p and q are propositions. The conjunction of p and q, denoted by p \(\wedge\) q is the proposition “p and q”. The conjunction p \(\wedge\) q is true when both p and q are true and is false otherwise.
|p||q||p \(\wedge\) q|
The disjunction of p and q, denoted by p \(\vee\) q, is the proposition p or q. The disjunction p and q is false when both p and q are false and is true otherwise.
|p||q||p \(\vee\) q|
Exclusive or of two Propositions
Let p and q be two propositions. The exclusive or of p and q , denoted by p \(\oplus\) q, is the proposition that is true when exactly one of p and q is true and is false otherwise.
|p||q||p \(\oplus\) q|
The conditional statement p \(\to\) q is the proposition if p then q. p \(\to\) q is false when p is true and q is false, and true otherwise. In this statement , p is called hypothesis and q is the conclusion. Also called implications. The statement p \(\to\) q is true when both p and q are true and when p is false.
|p||q||p \(\to\) q|
Consider a statement that a professor might make- “If you get 100% on the final,then you will get an A”.
Converse, Contrapositive, Inverse:
The proposition q \(\to\) p is called the converse of p \(\to\) q.
The contrapositive of p \(\to\) q is the proposition \(\lnot\)q \(\to\) \(\lnot\)p.
The proposition \(\lnot\)p \(\to\) \(\lnot\)q is called the inverse of p \(\to\) q.
The biconditional statement p \(\leftrightarrow\) q is the proposition “p if and only if q”. The statement is true when p and q have the same truth values, and is false otherwise. This is also called bi-implications.
|p||q||p \(\leftrightarrow\) q|
|p||q||\(\lnot\)q||p \(\vee\) \(\lnot\)q||p \(\wedge\) q||(p \(\vee\) \(\lnot\)q) \(\to\) (p \(\wedge\) q)|
|p||q||p \(\vee\) q||\(\lnot\)(p \(\vee\) q)||\(\lnot\)q||\(\lnot\) q||\(\lnot\)p \(\wedge\) \(\lnot\)q|
A Tautology is a formula which is always true for every value of its propositional variables.
|A||B||A \(\to\) B||(A \(\to\) B) \(\wedge\) A||[(A \(\to\) B) \(\wedge\) A] \(\to\) B|
A Contradiction is a formula which is always false for every value of its propositional variables.
|A||B||A \(\vee\) B||\(\lnot\)A||\(\lnot\) B||(\(\lnot\)A) \(\wedge\) (\(\lnot\)B)||(A \(\vee\) B) \(\wedge\) [(\(\lnot\)A) \(\wedge\) (\(\lnot\) B)]|
A Contingency is a formula which has both some true and some false values for every value of its propositional variables.
|A||B||A \(\vee\) B||\(\lnot\)A||(A \(\vee\) B) \(\wedge\) (\(\lnot\) A)|
Two statements X and Y are logically equivalent if any of the following two conditions −
|A||B||A \(\vee\) B||\(\lnot\)(A \(\vee\) B)||\(\lnot\)A||\(\lnot\)B||[(\(\lnot\)A) \(\wedge\) (\(\lnot\) B)]|
|A||B||\(\lnot\) (A \(\vee\) B)||[(\(\lnot\)A) \(\wedge\) (\(\lnot\)B)]||[\(\lnot\) (A \(\vee\) B)] \(\Leftrightarrow\) [(\(\lnot\)A) \(\wedge\) (\(\lnot\)B)]|