lecture 7, August 2, 2018 .pdf

# lecture 7, August 2, 2018 .pdf - A little more logic and...

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A little more logic, and functions from set theoretic viewpoint Jacob Shapiro MATH1115 Lecture 7, August 2, 2018
Proof by cases Example Prove that for every x Z , the integer x ( x + 3) is even. Proof. Since x Z , there are two possibilities, either x is even or x is odd. If x is even, then x = 2 k for some k Z , and we get x ( x + 3) = 2 k (2 k + 3) = 2( k (2 k + 3)) . Because k (2 k + 3) Z , this says that x ( x + 3) is even. On the other hand, if x is odd, then x = 2 k 0 + 1 for some k 0 Z , and then x ( x + 3) = (2 k 0 + 1)(2 k 0 + 4) = 2(2 k 0 + 1)( k 0 + 2) . Since (2 k 0 + 1)( k 0 + 2) Z , we conclude that x ( x + 3) is even.
Cartesian product of sets Definition Let A and B sets. We define the set A × B , called the Cartesian product of A and B , to be the set of ordered pairs ( x , y ) such that x A and y B . That is, A × B = { ( x , y ) : x A and y B } .
Cartesian products Examples I R 2 = R × R = { ( x , y ) : x R and y R } . I R × (0 , ) = { ( x , y ) : x R , y R and y > 0 } I { 2 } × [ - 5 , 3) = { (2 , x ) : x R and - 5 x < 3 } I R 3 = R × R × R = { ( x , y , z ) : x , y , z R } I If A 6 = , then A × ∅ =?
Cartesian products Examples I R 2 = R × R = { ( x , y ) : x R and y R } . I R × (0 , ) = { ( x , y ) : x R , y R and y > 0 } I { 2 } × [ - 5 , 3) = { (2 , x ) : x R and - 5 x < 3 } I R 3 = R × R × R = { ( x , y , z ) : x , y , z R } I If A 6 = , then A × ∅ =? Answer: A × ∅ = .
Informal notion of a function Usually we think of a function f as some kind of rule. The function has a particular domain , perhaps R , a closed interval [ a , b ], or something else. Given a point x in the domain, we associate to x another number f ( x ) R . Examples I f : [0 , 1] R , f ( x ) = x 2 , I f : R ( - π 2 , π 2 ) , f ( x ) = arctan( x ) . Thinking about a function in this manner will be sufficient for most of the proofs we do in this course.
Formal definition of a function Definition A function f consists of three things I a set A called the domain of f , I a set B called the codomain of f , I a set S A × B so that x A , a unique y B such that ( x , y ) S . In this case we write f : A B . For each x A , we refer to its corresponding y B as f ( x ). Example I Formally speaking, f : [0 , 1] R , f ( x ) = x 2 is the subset { ( x , x 2 ) : x R } ⊆ [0 , 1] × R (often called the graph of f ).
Formal definition of a function Definition A function f consists of three things I a set A called the domain of f , I a set B called the codomain of f , I a set S A × B so that x A , a unique y

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