Functions

Functions - CSE 1400 Applied Discrete Mathematics Functions...

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CSE 1400 Applied Discrete Mathematics Functions Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Functions 1 Counting Functions 2 Onto Functions 3 One-to-One Functions 3 Cardinality Deﬁned 4 Inverse Function 4 Function Composition 4 Problems on Functions 5 Abstract A function is a deterministic relation. Functions In college algebra and pre-calculus, you have studied functions such as y = x y = x 2 y = log x y = e x y = cos x y = sin x In college algebra, the function concept is often explained by the vertical line test: “If every vertical line crosses a graph at most once, the graph represents a function.” A given value x in the domain of a function maps to one and only one value y in the range of the function. Functions are denoted by writing f : X Y where X is the domain and Y is the co-domain of function f . The range G of f is the subset of Y for which there is an x X such that y = f ( x ) . The range of f also called the image of X and written G = f ( X ) Y To write f is a function from X to Y in predicate logic requires the conjunction of two propositions. ( x X )( y Y )( f ( x ) = y )

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cse 1400 applied discrete mathematics functions 2 x = 1/2, y = 3/2 x x = 1/2, y = - 3/2 y - 1 - 1 2 1 - 1 - 1 2 1 2 1 x 2 + y 2 = 1 is relation, but not a function. Figure 1 : The vertical line x = 0.5 crosses the circle twice. The relation y = ± 1 - x 2 between x and y is not a function. and ( x X )( y , z Y )(( f ( x ) = y f ( x ) = z ) = y = z ) The ﬁrst proposition states that every element x in the domain maps to some element y in the range : The function is total . The second predicate states that if x maps to two different names y and z , their values are, in fact, equal. There are several things worth knowing about functions. For instance, it is useful to know The number of different functions from X to Y that can be de- ﬁned. Basic properties such as
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Functions - CSE 1400 Applied Discrete Mathematics Functions...

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