CSNB143 – Discrete Structure

1. CSNB143 – Discrete Structure LOGIC

2. LOGIC Learning Outcomes • Student should be able to know what is it means by statement. • Students should be able to identify its connectives and compound statements. • Students should be able to use the Truth Table without difficulties.

3. LOGIC • Statement or proposition is a declarative sentence with the value of true or false but not both. • Ex 1: Which one is a statement? • The world is round. • 2 + 3 = 5 • Have you taken your lunch? • 3 - x = 5 • The temperature on the surface of Mars is 800F. • Tomorrow is a bright day. • Read this!

4. LOGIC Logical Connectives and Compound Statements • Statement usually will be replaced by variables such as p, q, r or s. • Ex 2: p: The sun will shine today. q: It is a cold weather. • Statements can be combined by logical connectives to obtain compound statements. • Ex 3: AND (p and q): The sun will shine today and it is a cold weather. • Connectives AND is what we called conjunction for p and q, written p  q. The compound statement is true if both statements are true. • Connectives OR is what we called disjunction for p and q, written p  q. The compound statement is false if both statements are false.

5. LOGIC • To prove the value of any statement (or compound statements), we need to use the Truth Table.

6. LOGIC Negation for any statement p is not p. written as ~p or p. The Truth Table for negation is: Ex 4: Find the value of (~pq)  p using Truth Table. Note: Always start with p, followed by q, then r, then s.

7. LOGIC Conditional Statements • If p and q are statements, the compound statement if p then q, denoted by p  q is called a conditional statement or implication. • Statement p is called the antecedent or hypothesis; and statement q is called consequent or conclusion. The connective if … then is denoted by the symbol . • Ex 5: a) p : I am hungry q : I will eat b) p : It is cold q : 3 + 5 = 8 • The implication would be: a) If I am hungry, then I will eat. b) If it is cold, then 3 + 5 = 8. • Take note that, in our daily lives, Ex 5 b) has no connection between statements p and q, that is, statement p has no effect on statement q. However, in logic, this is acceptable. It shows that, in logic, we use conditional statements in a more general sense.

8. LOGIC Its Truth Table is as below: To understand, use: p = It is raining q = I used umbrella • If p  q is an implications, then the converse of it is the implication q  p, and the contrapositive of it is ~q  ~p. • Ex 6: p = It is raining q = I get wet Get its: 1) converse 2) contrapositive.

9. LOGIC • If p and q are statements, compound statement p if and only if q, denoted by p  q, is called an equivalence or biconditional. • Its Truth Table is as below: To understand, use: p = 3 > 2 q = 0 < 3-2 • Notis that p  q is True in two conditions: both p and q are True, or both p and q are false. • Another meaning that use the symbol  includes: • p is necessary and sufficient for q • if p, then q, and conversely

10. LOGIC • In general, compound statement may contain few parts in which each one of it is yet a statement too. • Ex 7: Find the truth value for the statement (p  q)  (~q  ~p) • A statement that is true for all possible values of its propositional variables is called a tautology. • A statement that is always false for all possible values of its propositional variables is called a contradiction. • A statement that can be either true or false, depending on the truth values of its propositional variables is called a contingency

11. LOGIC Logically Equivalent • Two statements p and q are said to be logically equivalent if p  q is a tautology. Ex 8: Show that statements p  q and (~p)  q are logically equivalent (from previous example). Contradiction. Ex 9: p  ~p Contingency Ex10: p  q (p  q) • p  q p  q p  q (p  q) T T ? F T T F

12. LOGIC Quantifier • Quantifier is used to define about all elements that have something in common. • Such as in set, one way of writing it is {x | P(x)} where P(x) is called predicate or propositional function, in which each choice of x will produces a proposition P(x) that is either true or false. • A= {x | x is an integer less than 8} . P(1) is true, 1  A. Choice of x lead to P(x) is true. • There are two types of quantifier being used: • Universal Quantification () of a predicate P(x) is the statement “For all values of x, P(x) is true” In other words: • for every x • every x • for any x

13. LOGIC Quantifier • Existential Quantification () of a predicate P(x) is the statement “There exists a value of x for which P(x) is true” In other words: • there is an x • there is some x • there exists an x • there is at least one x • Ex 9: Q(x) : x + 1 < 4. False or True for Universal and Existential Quantifier.