Definition Definition Set and element Cardinality of a set Sets Sets Definition Definition Empty set (null set) Finite and infinite sets Sets Sets Definition Definition Ordered pair / n-tuple Cartesian product Sets Sets Definition Definition Subset Proper subset Sets Sets Definition Definition Power set Partition Sets Sets The cardinality of a set is the number of elements in A set is a collection of objects; the objects in the set the set. are called elements. The cardinality of the set A is denoted by jAj or #A. If A is a set, and a is an element of A, then we write a 2 A. A set is finite if its cardinality is finite, that is, the set contains finitely many elements. The empty set is the set that contains zero elements. A set is infinite if it contains infinitely many The empty set is denoted by ?. elements. If A and B are sets, the cartesian product A × B is the set of ordered pairs where the first element is An ordered pair is an ordered list of two elements. from A, and the second is from B. More generally, an ordered n-tuple is an ordered list A × B = f(a; b): a 2 A; b 2 Bg: of n elements. The standard notation is to use a comma separated list enclosed by parenthesis: (a1; a2; : : : ; an). (a; b) 2 A × B , a 2 A and b 2 B The set B is a proper subset of A if B is a subset of A set A is a subset of a set B if every element of A is A that is not equal A, and we write B ⊂ A. an element of B. B ⊂ A , B ⊆ A and B 6= A A ⊆ B , x 2 A ) x 2 B Let A be a set, and let P ⊆ P(A). The set P is a The power set of a set A is the set of all subsets of A. partition of A if The power set of A is denoted by P(A). 1. S X = A; X2P P(A) = fB : B ⊆ Ag 2. if X1;X2 2 P , then X1 \ X2 = ? , X1 6= X2. Definition Definition Set equality Union Sets Sets Definition Definition Finite and infinite union Intersection Sets Sets Definition Definition Finite and infinite intersection Set difference Sets Sets Definition Theorem Disjoint sets Double containment principle Sets Sets Definition Theorem Complement De Morgan's laws Sets Sets The union of two sets, A and B, is the set of all element in A or in B. The union of these sets is denoted by A [ B. Two sets, A and B, are equal if all the elements of A are elements of B and vice versa. A [ B = fx : x 2 A or x 2 Bg A = B , x 2 A if and only if x 2 B x 2 A [ B , x 2 A or x 2 B A finite union is the union of finitely many sets. An The intersection of two sets, A and B, is the set of all infinite union is the union of infinitely many sets. element in A and in B. Let A1;A2;A3;::: be sets, then The intersection of these sets is denoted by A \ B. n [ A \ B = fx : x 2 A and x 2 Bg Ai = fx : x 2 Ai for some 1 ≤ i ≤ ng i=1 [ x 2 A \ B , x 2 A and x 2 B Ai = fx : x 2 Ai for some i 2 Ng i2N A finite intersection is the intersection of finitely many sets. An infinite intersection is the intersection If A and B are sets, the, is the difference A − B is the of infinitely many sets. set of elements in A that are not in B. Let A1;A2;A3;::: be sets, then A − B = fx : x 2 A and x2 = Bg n \ Ai = fx : x 2 Ai for all 1 ≤ i ≤ ng i=1 x 2 A − B , x 2 A and x 62 B \ Ai = fx : x 2 Ai for all i 2 Ng i2N Let A and B be sets. Then A = B if and only if A ⊆ B and B ⊆ A. Two sets, A and B, are disjoint if A \ B = ?. A = B , A ⊆ B and B ⊆ A The complement of a set A is the set of all elements that are not in A, and is denoted by Ac or A. For any sets A and B, If A ⊆ B, then the complement of A in B is the set of elements in B that are not in A, i.e. Ac = B − A. (A \ B)c = Ac [ Bc; (A [ B)c = Ac \ Bc: Ac = fx : x2 = Ag x 2 Ac , x 62 A Definition Definition Statement Logical statement Logic Logic Definition Definition Logical and Logical or Logic Logic Definition Definition Logical negation Implies Logic Logic Definition Definition Converse If and only if Logic Logic Definition Definition Contrapositive Universal quantifier: for all Logic Logic A statement is a sentence or mathematical expression A statement is a sentence or mathematical expression that is definitely true or definitely false. that is definitively true or definitively false. The statement \P or Q" is true if P is true or Q is The statement \P and Q" is true if both P is true true (or both statements are true). The statement \P and Q is true. Otherwise \P and Q" is false. and Q" is false only if both P is false and Q is false. P ^ Q is true , P is true and Q is P _ Q is true , P is true or Q is true true The negation of a statement P is the statement :P . The statement \P implies Q"(P ) Q) is false if P is The statement :P is true if P is true. The statement true and Q is false. Otherwise the statement is true. of :P is false if P is true. P ) Q is false , P is false and Q is P is true (resp. false) ,:P is false (resp. true true) The statement \P if and only if Q"(P , Q) is equivalent to the statement (P ) Q) ^ (Q ) P ). In The converse of P ) Q is the statement Q ) P . In other words, P , Q is true if both P ) Q and general, these two statements are independent, Q ) P are true. meaning that the truthfulness of one statement does P , Q , (P ) Q) ^ (Q ) P ) not determine the truthfulness of the other. The for all/each/every/any statement takes the The contrapositive of the statement \if P , then Q" is form: \for all P , we have Q." In other words, Q is the statement \if :Q, then :P ". These statements true whenever P is true. In this light, \for all" are equivalent, meaning that they are either both statements can often be reworded as \if-then" true or both false. statements (and vice versa). P ) Q ,:Q ):P 8P , we have Q , P ) Q Definition Negation of P ^ Q Existential quantifier: there exists Logic Logic Negation of P _ Q Negation of P ) Q Logic Logic Negation of 8P; we have Q Negation of 9P such that Q Logic Logic Definition Definition Theorem Proof Logic Logic Definition Definition Definition List and entries Logic Counting The there exists statement takes the form: \there exists P such that Q." This statement is true if there is at least one case where P is true and Q is true. (It maybe that there are many cases where P is false but :(P ^ Q) = :P _:Q Q is true.) 9P such that Q , it is sometimes the case that P ) Q :(P ) Q) = P ^ :Q :(P _ Q) = :P ^ :Q :(9P; such that Q) = 8P we have :Q :(8P; we have Q) = 9P; such that :Q A proof of a theorem is a written verification that A theorem is a mathematical statement that is true shows that the theorem is definitely and and can be (and has been) verified as true. unequivocally true. A list is an ordered sequence of objects. The objects A definition is an exact, unambiguous explanation of in the list are called entries. Unlike sets, the order of the meaning of a mathematical word or phrase. entries matters, and entries may be repeated. Definition Definition List length List equality Counting Counting Definition Theorem Empty list Multiplication principle Counting Counting Definition Definition Factorial n choose k Counting Counting Theorem Theorem Binomial theorem Inclusion-exclusion Counting Counting Theorem Definition Addition principle Even Counting Counting Two lists L and M are equal if they have the same The length of a list is the number of entries in the length, and the i-th entry of L is the i-th entry of M. list. Suppose in making a list of length n there are ai possible choices for the i-th entry. Then the total The empty list is the list with no entries, and is number of different lists that can be made in this way denoted by (). is a1a2a3 ··· an. If n and k are integers, and 0 ≤ k ≤ n, then If n is a non-negative integer, then n! is the number of non-repetitive lists of length n that can be made n n! = : from n symbols. Thus 0! = 1, and if n > 1, then n! is k k!(n − k)! the product of all integers from 1 to n. That is, if n > 1, then n! = n(n − 1)(n − 2) ··· 2 · 1.