PROPOSITIONAL CALCULUS A proposition is a complete declarative sentence that is either TRUE (truth value T or 1) or FALSE (truth value F or 0), but not both.
These are not propositions!
Connectives and Compound Propositions
PROPOSITIONAL CONNECTIVES
Negation of a Proposition The negation of a proposition P is denoted by P and is read as not P. The truth table is shown below:
Example 1: Negation of a Proposition 1. P: It will rain today. P: It will not rain today. 2. Q: Angela is hardworking. Q: Angela is not hardworking. 3. R: You will pass this course. R: You will not pass this course.
Conjunction of Propositions The proposition P and Q, denoted by P Q, is called the conjunction of P and Q. Other keywords: but, nevertheless
Conjunction of Propositions The truth table is shown as follows:
Example 2: Conjunction of a Proposition 1. P: Sofia is beautiful. Q: Anton is strong. P Q: Sofia is beautiful and Anton is strong. 2. S: The stock exchange is down. T: It will continue to decrease. S T: The stock exchange is down and it will continue to decrease.
Disjunction: Inclusive or The proposition P or Q, denoted by P Q, is called the disjunction of P or Q. This is also referred to as the inclusive or. Other keywords: unless
Disjunction: Inclusive or The truth table is shown below:
Example 3: Inclusive or 1. P: This lesson is interesting. Q: The lesson is easy. P Q: This lesson is interesting or it is easy. 2. S: I want to take a diet. T: The food is irresistible. S T: I want to take a diet or the food is irresistible.
Disjunction: Exclusive or The disjunction proposition P or Q but not both, denoted by P Q, is called the exclusive or. The truth table is shown as follows:
Example 4: Exclusive or P: Presidential candidate A wins. Q: Presidential candidate B wins. P Q: Either presidential candidate A or B wins.
Implications or Conditionals The proposition If P, then Q, denoted by P Q is called an implication or a conditional. Equivalent propositions: P only if Q, Q follows from P, P is a sufficient condition for Q, Q whenever P
Implications or Conditionals The truth table for an implication is shown as follows:
Example 5: Implications P: It is raining very hard today. Q: Classes are suspended. P Q: If it is raining very hard today, then classes are suspended.
Related Implication: Converse The converse of the proposition If P, then Q is the proposition If Q, then P. In symbols, the converse of P Q is Q P. Example: The converse of the proposition P Q: If it is raining very hard today, then classes are suspended. is the proposition Q P: If classes are suspended, then it is raining very hard today.
Related Implication: Contrapositive The contrapositive of the proposition If P, then Q is the proposition If not Q, then not P. In symbols, the contrapositive of P Q is Q P. Example: The contrapositive of the proposition P Q: If it is raining very hard today, then classes are suspended. is the proposition Q P: If classes are not suspended, then it is not raining very hard today.
Related Implication: Inverse The inverse of the proposition If P, then Q is the proposition If not P, then not Q. In symbols, the inverse of P Q is P Q. Example: The inverse of the proposition P Q: If it is raining very hard today, then classes are suspended. is the proposition Q P: If it is not raining very hard today, then classes are not suspended.
Biconditionals The proposition P if and only if Q, denoted by P Q is called a biconditional. Equivalent propositions: P is equivalent to Q, P is a necessary and sufficient condition for Q
Biconditionals The truth table for a biconditional is shown as follows:
Example 6: Biconditionals P: I will pass Matapre. Q: My grade is at least 60. P Q: I will pass Matapre if and only if my grade is at least 60.
TRUTH TABLE SUMMARY
Truth Tables In constructing a truth table, the number of rows is equal to 2 n where n is the number of propositional variables. For example, if there are 4 propositional variables, then the truth table will consist of 2 4 = 16.
Assignment of Values For two propositional variables, we have 4 rows for the truth table and the assignment of values are shown as follows:
Assignment of Values For three propositional variables, we have 8 rows:
Assignment of values
Types of Propositional Forms There are three types of propositional forms: Tautology Contradiction Contingency
Tautology A propositional form that is true under all circumstances is called a tautology. Example: The proposition P Q P Q is a tautology. The truth table is shown as follows:
Tautology
Contradiction A propositional form that is false under all circumstances is called a contradiction. Example: The proposition P Q (Q P) is a contradiction. The truth table is shown as follows:
Contradiction
Contingency A propositional form that is neither a tautology nor a contradiction is called a contingency. Example: The proposition (P Q) R is a contingency. The truth table is shown as follows:
Contingency
Knowledge Check 1.3 Test 1: Determine if the following propositional forms is a tautology, contradiction or contingency by constructing truth tables for each. 1. P P 2. Q S P 3. Q P P Q 4. Q S P S Test 2: Turn on page 15 and answer number 18.
Logically Equivalent Propositions
Some Logically Equivalent Propositions
Some Logically Equivalent Propositions
Remark: Implications
When is a Mathematical Reasoning Correct?
Rule of Inference: Addition 1. Addition P P Q In this rule of inference, we can add any propositional variable to another with the use of the logical connective or.
Rule of Inference: Simplification 2. Simplification P Q P Illustration: Give the appropriate conclusion using simplification rule. (R S) T?
Rule of Inference: Conjunction 3. Conjunction P Q P Q Illustration: Give the appropriate conclusion using conjunction rule. S T N R?
Rule of Inference: Modus Ponens 4. Modus Ponens P Q P Q Illustration: Give the appropriate conclusion using modus ponens rule. (M N) T M N?
Rule of Inference: Modus Tollens 5. Modus Tollens P Q Q P Illustration: Give the appropriate conclusion using modus tollens rule. M (B C) (B C)?
Rule of Inference: Disjunctive Syllogism 6. Disjunctive Syllogism P Q P Q Illustration: Give the appropriate conclusion using disjunctive syllogism rule. Z (X Y) Z?
Rule of Inference: Hypothetical Syllogism 7. Hypothetical Syllogism P Q Q R P R Illustration: Give the appropriate conclusion using hypothetical syllogism rule. (X N) T T (A B)?
Some Applications
Some Applications
Knowledge Check 1.4 State the rule of inference by which the conclusion follows from its premise/s. 1. A (B C) B C A D 2. T U W D [T U W ] P
Knowledge Check 1.4 3. E F G H E F G H 4. I J K L I J K L
Knowledge Check 1.4 5. M N N Q (M N) ( N Q) 6. P Q R S P Q R S