Its all about connectivity, Lecture notes of Mathematics

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MODULE 2

Propositions &

Connectives

Introduction

Logic and mathematical reasoning have numerous applications in computer science. Rules in logic are used in the design of computer circuits, the development of computer programs, the verification of the correctness of programs, and many other ways.

Examples:

• All the ff: declarative sentences are propositions.
• 1. 8 is an even integer.
• 2. Kalibo is the capital of Aklan.
• 3. 6 (5) = 30.
• 4. √2 is an integer.

Propositions 1, 2, 3 are true, whereas 4 is false.

• Examples of sentences that are not propositions:
• 1. Why are you here?
• 2. Be attentive in class.
• 3. a – 5 = 10
• 4. m + n = p

An acceptable proposition is given the decision value true (or 1 ) , while an unacceptable statement is assigned a decision value false (or 0 ). An array of decision value (truth value) is called a logical matrix (or truth table).

Logical operators

• Many mathematical statements are constructed by combining one or more propositions. These new propositions are formed using logical operators. The logical operators that are used to form new propositions from 2 or more existing propositions are called connectives.

Statement or Proposition P ^ Q True if and only if P and Q are both true P v Q True if and only if P is true or Q is true or both are true P Q True under all circumstances except when P is true and Q is false. P Q True if and only if P and Q are both true or both false ➢ Must express a complete thought ➢ A declarative sentence or statement that is either true or false but not both.

Negation

• Definition: let p be a proposition. The statement “ It is not the case that p” is another proposition, called the negation of p , denoted by ¬p. the proposition ¬p is read “not p”.

• The truth value of a

true proposition is T or 1.

The truth value of a false

proposition is F or 0.

• A truth value or truth

matrix displays the

relationships between

the truth values of

propositions.

The matrix gives 2 possible decisions for q, since a proposition may either be T or F. Table 1: The Truth Matrix of the Negation of a Proposition p q 1 0

Examples:

• 1 ) Consider the following sentences:
• p: 6 is an even integer.
• q: 7 is an odd integer.
• The conjunction of this proposition, p ٨ q , is the proposition “ 6 is an even integer & 7 is an odd integer.”

• 2 ) Let p: 4 divides 16.
• q: 4 divides 24.
• p ٨ q , is the proposition,

“ 4 divides 16 and 4 divides

• or “ 4 divides both 16 and

Table 2: The Truth Matrix for the Conjunction of the 2 Propositions

p q p ٨ q

• Note:

• There is only one

condition for p ٨ q to

be true & this is when

both statements are

true.