Summary: Mathematical Proofs A Transition To Advanced Mathematics | 9780134746753 | Gary Chartrand, et al

Read the summary and the most important questions on Mathematical Proofs A Transition to Advanced Mathematics | 9780134746753 | Gary Chartrand; Albert D. Polimeni; Ping Zhang

• 1.1 Describing a Set

• What is a set?

  It is a small or large regrouping of elements containing similar properties.
  A set is finite if we can count the number of elements it contains, |S|=n, for n in N. Otherwise a set is infinite
  S={1,2,3...} set of natural numbers
  S:{(2,5)}

• 1.2 Subsets

• What is a subset?

  A set A is a subset of B if it is contained in B, for every x in A, x is also in B.

• What is a Power Set?

  It is the set consisting of all subsets of a given set, A. It is denoted by P(A)

• 1.3 Set Operations

• What are some Set operations?

  Union: it is the operation that allows an element to be in set A or set B, AUB
  
  Intersection: it is the operation that allows an element to be in set A and set B, AnB
  If two sets don't share any elements, they are called disjoint
  
  Difference: is the operation where we are looking for element that are in A but not in B, A\B
  
  Complement: is the operation where we are looking for all elements outside of given set A,

• 1.5 Partitions of Sets

• What is a partition?

  A partition of a set A, is a collection S of nonempty subsets of A such that every element in A belongs to only one subset of A

• 1.6 Cartesian Products of Sets

• What is a Cartesian product?

  It is the set consisting of all ordered pairs of elements in which the fist coordinate belongs to set A and the second belongs to set B.
  AxB : {(a1,b1), (a2,b2)...,(an,bn)} where a1,... Is in A and b1,... Is in B

• 2.2 Negations

• What is the negation of a statement P?

  Not P

• What is the law of excluded middle?

  It says that a statement is either true or it's negation is true.
  Pv(~P)

• 2.3 Disjunctions & Conjunctions

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• What is a disjunction?

  If P and Q are statements, a disjunction of P and Q is the statement P or Q, denoted PvQ
  In a disjunction one one P or Q must be true for PvQ to be true
  Negation of disjunction: ~(PvQ)= (~P)v(~Q)
  Distributivity : Pv(Q^R) <-> (PvQ)^(PvR)

• 2.5 More Implications

• What are other ways to formulate an implication?

  If P then Q
  Q if P 
  P implies Q
  Q is necessary for P
  Q is sufficient for P
  
  In P->Q, P is called the premise/hypothesis and Q is calle dthe conclusion