Teacher Service Commission, Mathematics, Secondary Level Symbolic logic: Logical connectives, algebra of propositions Logic is study of mathematical reasoning. It is also called proportional calculus. It is also called calculus of statement. Mathematical Sentence A sentence expressing mathematical object or relation or facts is called mathematical sentence. For example 1. 2 + 3 = 5 2. 2 + 3 = 6 3. 2 + x = 5 The statement 2 + 3 = 5 is true The statement 2 + 3 = 6 is false The statement 2 + x = 5 may be true or false depending on the value of x. If x = 1 then 2 + 1 = 5 is false. If x = 3 then 2 + 3 = 5 is true. Thus, we can say that the mathematical sentence containing the variable becomes either true or false depending upon the value of the variable. There are two types of mathematical sentence. 1. Open Sentence 2. Declarative Sentence Open Sentence The sentence which is true or false but not sure, is called open sentence. For example, 2 + x = 5 is open sentence. It can be true or false. Note Open sentence has two truth value, T or F. (True or False)

1

Teacher Service Commission, Mathematics, Secondary Level Declarative Sentence The sentence which are either true of false but not both, is called declarative sentence. For example, 2 + 3 = 5 is true sentence. It can be true but not false. Note Open sentence has only one truth value, Either T or F but not both. The declarative sentence are called mathematical statement. There are two types of mathematical statement. 1. Simple statement 2. Compound statement Simple statement A simple statement is a statement which has one subject and one predicate. For example, the statement: Kathmandu is the capital of Nepal is a simple statement. Kathmandu is the subject and is the capital of Nepal is the predicate. Simple statement is basic building block of logic. Such simple statement is either true or false. Simple is simple. The truth value of simple depends on the statement itself, not to any other statement. Compound statement A compound statement is a statement which has two or more subject and their predicate. For example, the statement: Today is Sunday and Today is Cloudy 2

Teacher Service Commission, Mathematics, Secondary Level is a compound statement. Such compound statement are combination of two or more simple statements. Thus the truth value of compound statement depends on one or more its connecting statements. Logical Connectives दईु वा दईु भन्दा बढी simple statements हरुलाई जोडेर compound statement बनाउने “operation” लाई logical connectives भननन्छ । यसलाई logical operations पनन भननन्छ।

Logic मा यस्ता logical operations हरु मख् ु यतः पााँचवटा छन । 1. Conjunction 2. Disjunction 3. Negation 4. Conditional 5. Bi-conditional Conjunction In logic, a conjunction is a compound statement formed by using the word and to join two simple statements. The symbol for this is Λ. (whenever you see Λ read 'and') When two simple statements, p and q, are joined in a conjunction statement, the conjunction is expressed symbolically as p Λ q. Example Simple Sentences p: Ram eats eggs. q: Maria drinks coca. The truth table 3

Compound Sentence: conjunction p Λ q : Ram eats eggs, and Maria drinks coca.

Teacher Service Commission, Mathematics, Secondary Level

Disjunction In logic, a disjunction is a compound statement formed by using the word or to join two simple statements. The symbol for this is ν. (whenever you see ν read 'or') When two simple statements, p and q, are joined in a disjunction statement, the disjunction is expressed symbolically as p ν q. Simple Sentences p: The clock is slow. q: The time is correct.

Compound Sentence: disjunction p ν q : The clock is slow, or the time is correct.

The truth table

Negation In logic, a negation is a compound statement formed by using the word not. The symbol to indicate negation is : ~ Indicates the opposite.

4

Teacher Service Commission, Mathematics, Secondary Level Original Statement Today is Monday. That was fun.

Negation of statement Today is not Monday. That was not fun.

The truth table

Conditional In logic, a conditional statement is compound sentence that is usually expressed with the key words 'If....then...'. Using the variables p and q to represent two simple sentences, the conditional "If p then q" is expressed symbolically as p  q. Simple Sentences p: You are absent q: You have a make up assignment to complete. Compound Sentence: Conditional p  q: If you are absent, then you have a make up assignment to complete. The truth table

5

Teacher Service Commission, Mathematics, Secondary Level Converse, Inverse and Contrapositive of Conditional Statement Let p  q be a conditional statement, then three other conditional statements can be formed. There are called Converse, Inverse and Contrapositive of given Conditional Statement. Conditional Statement

Converse

Inverse

Contrapositive

pq

 p  q

Example

6

q p

q   p

Teacher Service Commission, Mathematics, Secondary Level Truth Table of Fundamental Operations Logical Equivalence Tautology Contradiction Algebra of Logics (Propositions) 1. Idempotent law p p  p 2. Commutative law pq  q p 3. Associative Law  p  q  r  p  q  r  4. Distributive Law p  q  r    p  q   p  q ( p  q )   p   q 5. De Morgan’s Law

Exercise Find the inverse, converse, and contrapositive of the statement. 1. If you visited Dallas, then you’ve been to Texas. Converse: ______________________________________ Inverse: ________________________________________ Contrapositive: ________________________________ 2. If freezing rain is falling, then the air temperature is 32°F or less. Converse: ______________________________________ Inverse: ________________________________________ Contrapositive: ________________________________ 3 If two planes intersect, then their intersection is a line. Converse: ______________________________________ Inverse: ________________________________________ Contrapositive: _________________________

7