tailieunhanh - Lecture Discrete structures: Chapter 3 - Amer Rasheed

This chapter presents a justification for a mathematical framework, the ceiling and floor functions, L’Hôpital’s rule, logarithms, arithmetic and other polynomial series, geometric series, recurrence relations, weighted averages, combinations. | (CSC 102) Lecture 3 Discrete Structures Previous Lecture Summary Logical Equivalences. De Morgan’s laws. Tautologies and Contradictions. Laws of Logic. Logical Equivalences using Logical Laws. Lecture`s outline Conditional Propositions. Negation, Inverse and Converse of the conditional statements. Contra positive . Bi conditional statements. Necessary and Sufficient Conditions. Conditional statements and their Logical equivalences. Conditional propositions Definition If p and q are propositions, the conditional of q by p is if p then q or p implies q and is denoted by p→q. It is false when p is true and q is false otherwise it is true. Examples If you work hard then you will succeed. If John lives in Islamabad, then he lives in Pakistan. Implication (if - then) Binary Operator, Symbol: P Q P Q T T T T F F F T T F F T Examples “The online user is sent a notification of a link error if the network link is down”. The statement is equivalent to “If the network link is down, then the online user is sent a notification of a link error.” Using p : The network link is down, q : the online user is sent a notification of a link error. The statement becomes (q if p) p → q. Interpreting Conditional Statements Examples “When you study the theory, you understand the material”. The statement is equivalent to (using if for ‘‘when’’) “If you study the theory, then you understand the material.” Using p : you study the theory, q : you understand the material. The statement becomes (when p, q) p → q. Examples “Studying the theory is sufficient for solving the exercise”. The statement is equivalent to “If you study the theory, then you can solve the exercise.” Using p : you study the theory, q : you can solve the exercise. The statement becomes (p is sufficient for q) p → q. Other forms of conditional propositions if p and q are statements then “p only if q” means ‘’if p then q”. John will break the world`s record for the mile run only if he runs the mile in under four minutes. is | (CSC 102) Lecture 3 Discrete Structures Previous Lecture Summary Logical Equivalences. De Morgan’s laws. Tautologies and Contradictions. Laws of Logic. Logical Equivalences using Logical Laws. Lecture`s outline Conditional Propositions. Negation, Inverse and Converse of the conditional statements. Contra positive . Bi conditional statements. Necessary and Sufficient Conditions. Conditional statements and their Logical equivalences. Conditional propositions Definition If p and q are propositions, the conditional of q by p is if p then q or p implies q and is denoted by p→q. It is false when p is true and q is false otherwise it is true. Examples If you work hard then you will succeed. If John lives in Islamabad, then he lives in Pakistan. Implication (if - then) Binary Operator, Symbol: P Q P Q T T T T F F F T T F F T Examples “The online user is sent a notification of a link error if the network link is down”. The statement is equivalent to “If the network link is down, then the .