By Shankar, G. Rao
This article can be utilized by way of the scholars of arithmetic and computing device technology as an advent to the basics of discrete arithmetic. The ebook is designed based on the syllabi of b.e., b. Tech., mca and m.sc. (computer technology) prescribed in many of the universities of india. each one bankruptcy is supplemented with a few labored instance in addition to a few difficulties to be solved by way of the scholars. this may assist in a greater realizing of the topic. concerning the writer: g. Shanker rao has over 35 years of training adventure. He has been educating numerical research, actual research, operations learn and graph thought for the final 25 years. He used to be a member of many pro agencies. His parts of curiosity contain graph conception and computing device established arithmetic. almost immediately, he's operating as a member of employees, division of arithmetic, collage university of engineering, osmania college, hyderabad and used to be a former hod, division of arithmetic, girraj p.g. collage, nizamabad. desk of contents mathematical common sense set conception kin services and recurrence kinfolk boolean algebra common sense gates straight forward combinatorics graph thought algebraic buildings finite country machines
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Be two propositions. Then the following conditions are equivalent: 1. ) is a Tautology. 2. ) is a Contradiction. 3. ) is a Tautology. 1 holds. ) Example 1: (p ∧ q) ∧ ~(p ∨ q) is a contradiction. Hence p ∧ q ⇒ p ∨ q Example 2: (p → q) ∧ (q → r) → (p → r) is a tautology. ) is reflexive, anti-symmetric and transitive. Note: The symbols →, ⇒ are not the same ⇒ is not a connective nor P ⇒ Q is a statement formula (proposition). P ⇒ Q defines a relation in composite propositions P → Q. The symbol → is a connective and note that P → Q is just a proposition.
Contrapositive: If a steel rod does not stretch, then it has not been heated. 5. Converse: If x > 2, then x + 4 > 6 Inverse: If x + 4 > 6, then x > 2 6. p → ~q: It is cold, then it is not raining. q ↔ p: It is raining if and only if it is raining. p ↔ ~q: It is cold if and only is it is not raining. 7. p ∧ (q → r) ∧ (r → q) ∨ (r → p) ∧ (p → r) (b) q → ~p 14. (a) ~p ↔ ~q (d ) ~p ∧ q (c) ~p → ~q 16. (1) x ≠ 2 and x > ≠ 4 (2) x > 3 and x = 3 20. (a) False (b) True (c) True (d) False (e) True 21. (1) I like cats or I like dogs.
2 Notation Each of the objects in the set is called a member of an element of the set. The objects themselves can be almost anything. Books, cities, numbers, animals, flowers, etc. Elements of a set are usually denoted by lower-case letters. While sets are denoted by capital letters of English larguage. The symbol ∈ indicates the membership in a set. If “a is an element of the set A”, then we write a ∈ A. The symbol ∈ is read “is a member of ” or “is an element of ”. The symbol ∉ is used to indicate that an object is not in the given set.