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Statement - n Ans✔ A statement (proposition) is a sentence that is either true or false, but not both. Qs Negation - n Ans✔ A negation of a statement p is the statement "not p" or "it is not the case that p," and is denoted by ⌐p. A statement and its negation always have the opposite truth value. Qs Conjunction - n Ans✔ The conjunction of two statements p and q is the statement "p and q," and is denoted by p ᴧ q. The conjunction p ᴧ q is true if both p and q are true and is false otherwise. Qs Disjunction - n Ans✔ The disjunction of two statements p and q is the statement "p or q," and is denoted by p v q. The disjunction p v q is true if either of p and q are true or if both are true. Qs Conditional - n
Ans✔ A statement of the form "if p, then q" where p and q are statements, is called a conditional and is denoted by p -> q. Qs Forms of the conditional statement - n Ans✔ The conditional, p→ q can be stated in any of the following ways: If p, then q q if p p implies q p only if q p is sufficient for q q is necessary for p Qs Tautology - n Ans✔ A tautology is a statment that is always true no matter what truth values are assigned to the statements appearing in it. Qs Fallacy(or Contradiction) - n Ans✔ A fallacy(or Contradiction) is a statement that is always false. Qs Contingency - n Ans✔ A statement that is sometimes false and sometimes true is called a contingency
p v p ≡ p Qs Double Negation Law - n Ans✔ ⌐(⌐p) ≡ p Qs Negation Laws - n Ans✔ p ^ ⌐p ≡ F p v ⌐p ≡ T Qs Identity Laws - n Ans✔ p ᴧ T ≡ p p v F ≡ p Qs Universal Bound Laws - n Ans✔ p ᴧ F ≡ F p v T ≡ T Qs Absorption Laws - n Ans✔ P ᴧ ( p v q) ≡ p p v (p ᴧ q) ≡ p
Qs Variations of the Conditional Statement - n Ans✔ Conditional p→ q Converse q → p Inverse ⌐p → ⌐q Contrapositive ⌐q → ⌐p Qs Negation of Conditional - n Ans✔ ⌐( p → q) ≡ p ᴧ ⌐ q Qs Universe of Discourse - n Ans✔ The set of possible values Qs Universal Quantification - n Ans✔ The universal quantification of P(x) is the statement " For all x in the universe of discourse, P(x) is true." It is denoted by ¥ x P(x) Qs Existential Quantification - n Ans✔
p → q p
Therefore q Qs Modus Tollens - n Ans✔ p → q ⌐q
Therefore ⌐p Qs Reasoning by Transitivity(Law of Hypothetical Syllogism) - n Ans✔ p → q q → r
Therefore p → r Qs Disjunctive Syllogism - n Ans✔ p v q ⌐p
Therefore q Qs
Fallacy of the Converse - n Ans✔ p → q q
Therefore p Qs Fallacy of the Inverse - n Ans✔ p → q ⌐p
Therefore ⌐q Qs Even Number - n Ans✔ A even number is an integer that can be written in the form 2n, where n is an integer Qs Odd Number - n Ans✔ An odd number is an integer that can be written in the form 2n+1, where n is an integer Qs Rational Number - n Ans✔
Two sets A and B are said to be equal if they have the same elements. In this case we write A = B Qs Subset - n Ans✔ A set B is said to be a subset of a set A if every element of B is also an element of A. In other words, x Є B implies x Є A. To denote that B is a subset of A, we write B C A Qs Cardinality - n Ans✔ If A is the finite set, then the cardinality of A is the number of distinct elements in A, and is denoted by |A|. Qs Cartesian Product - n Ans✔ The Cartesian Product of two sets A and B is the set of all ordered pairs (a,b), where a Є A and b Є B. Is denoted by A x B. Therefore, A x B = { (a,b) | a Є A and b Є B}. Qs Power Set - n Ans✔ For any set S, the power set of S is the set whose elements are all the subsets of S, and it is denoted by P(S). in other words, P(S) = {B | B c S}. Qs
Union - n Ans✔ The union of two sets A and B is the set whose elements are elements of A, B, or both. It is denoted by A U B. Therefore, A U B= {x | x Є A or x Є B}. Qs Intersection - n Ans✔ The intersection of two sets A and B is the set whose elements are elements of both A and B. It is denoted by A ∩ B. Therefore, A ∩ B = { x | x Є A and x Є B}. Qs Difference - n Ans✔ Let A and B be sets. The diffence of A and B is the set whose elements are elements of A but not elements of B. It is denoted by A - B. Therefore, A - B = { x |x Є A and x Є B }. Qs Complement - n Ans✔ The complement of a set A is the set U - A whose elements are the elements of the universal set U that are not elements of A. It is denoted by Ā. Therefore, Ā = { x | x Ɇ A }. Qs Divides - n Ans✔
Ans✔ Given any integer a and positive integer d, there exists unique integers q and r such that a = dq + r with 0 ≤ r < d. Qs Greatest Common Divisor - n Ans✔ Let a and b be integers that are not both 0. The greatest common divisor of a and b, denoted gcd(a,b), is the largest integer d such that d | a and d | b. Qs Relatively Prime - n Ans✔ Two integers a and b are said to be relatively prime if gcd(a,b) = 1. Qs Lemma 3.1 - n Ans✔ If a, b , q and r are integers, and a = bq + r, then gcd(a,b) = gcd(b,r). Qs Euclidean Algorithm - n Ans✔ Used to find the greatest common divisor of a and b. It is the last nonzero number. Qs Theorem 3.6 - n Ans✔
If a and b are positive integers, then gcd(a,b) can be written as an integral linear combination of a and b. in other words, tehre exists integers s and t such that gcd(a,b) = sa + tb. Qs a modulo n (a mod n) - n Ans✔ If n is a positive integer and a is any integer, then the remainder we get when using the division algorithm to divide a by n is calld a modulo n. and is denoted a mod n Qs Congruent modulo n - n Ans✔ Let n be a positive integer. Two integers a and b are said to be congruent modulo n if n | (b - a). We write a ≡ b (mod n) to denote a is congruent to b modulo n, and write a ≠ b (mod n) to indicate that this is not the case. Qs Congruence Modulo n. - n Ans✔ Let n be a positive integer, and a and b any integers. Then a ≡ b (mod n) if and only if the leave the same remainder when divided by n using the division algorithm, that is if and only if a mod n = b mod n. Qs Properties of Modular Arithmetic - n Ans✔ Let a, b, c, d, and n be integers with n > 1, and suppose a ≡ b (mod n) and c ≡ d (mod n). Then,
- (a + c) ≡ (b + d) ( mod n)
- (a - c) ≡ ( b - d) ( mod n)
- ac ≡ bd (mod n)
Qs Domain - n Ans✔ Let f : A → B be a function from A to B. We call A the domain of f Qs Codomain - n Ans✔ Let f : A → B be a function from A to B. The set B is called the codomain of f. Qs Range - n Ans✔ Let f : A → B be a function from A to B. The set of all images of the elements of A under the function f is called the range of f. in other words range of f = { b Є B | b = f(a) for some a Є A }. Qs one-to-one - n Ans✔ A function f : A → B from A to B is called one-to-one if f (a1) = f (a2) implies a1 = a2 for all a1, a2 Є A. Qs Onto - n Ans✔ A function f : A → B from A to B is called onto if every element of B is an image of some element of A under f. That is, f is onto if given any b Є B, there exists some element a Є A such that
f (a) = b Qs one-to-one correspondence (bijection) - n Ans✔ A function f is called one-to-one correspondence (bijection) if it is both one-to-one and onto. Qs Composition - n Ans✔ Let g: A → B be a function from the set A to the set B, and let f : B → C be a function for the set B to the set C. Then the composition of f and g is the function f ᵒ g from A to C defined by ( f ᵒ g ) (a) = f ( g (a) ) Qs Inverse Function - n Ans✔ Let f : A → B be a one-to-one correspondence from the set A to the set B. Then the inverse function of f is the function f^-1: B → A defined by, f ^ -1 (b) = a, where a is the unique element of A for which f(a) = b. Qs Floor Function - n Ans✔ The floor function is a function from R to Z that maps x to the largest integer that is less than or equal to x. It is denoted by └x┘. Qs Ceiling Function - n
A set A is called countable(countably infinite) if there exists a bijection f : { 1,2,3,....} → A (i.e. - we can make an infinite lists of all elements of A.) Qs
