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Logical structure 2 show that the hypotheses it is

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LOGICAL STRUCTURE 2. Show that the hypotheses "It is not sunny this afternoon and it is colder than yesterday," "We will go swimming only if it is sunny," "If we do not go swimming, then we will take a canoe trip, " and " If we take a canoe trip, then we will be home by sunset" lead to the conclusion "We will be home by sunset. "
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FUNCTIONS
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FUNCTIONS Definition Let A and B be sets. A function f from A to B is an assignment of exactly one B element to each element of A . We write f(a)=b if b is the unique element of B assigned by the function f to the element a of A. If is a function from A to B , we write f:A→B .
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FUNCTIONS Definition If f is a function from A to B , we say that A is the domain of f and B is the codomain of f . If f(a)=b , we say that b is the image of a and a is a pre-image of b . The range of f is the set of all images of elements of A . Also, if f is a function from A to B , we say that f maps A to B .
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FUNCTIONS A function f from A to B is said to be one- to-one , or injective if for each b ϵ B, there is at most one a A with f(a)=b ϵ . A function is said to be an injection if it is one-to-one. A function f from A to B is called onto , or surjective if and only if for every element b B ϵ there is an a A ϵ element with f(a)=b . A function is said to be surjection if it is onto.
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FUNCTIONS The function f is a one-to-one correspondence , or bijection , if it is both one-to-one and onto. Example:Let S={1,2,3,4,5} and T={a,b,c,d}. For each question below if your answer is yes , give an example; if your answer is no , explain briefly. 1. Are there any one-to-one functions from S to T? 2. Are there any functions mapping S onto T?
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FUNCTIONS Definition(Inverse) Let f be a one-to-one correspondence from the set A to the set B . The inverse function of f is the function that assigns to an element b belonging to B the unique element a in A such that f(a)=b . The inverse function of f is denoted by f¯¹ . Hence, f¯¹ (b)=a when f(a)=b .
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FUNCTIONS Definition(Composition) Let g be a function from the set A to the set B and let f be a function from the set B to the set C . The composition of the functions f and g , denoted by f◦g is defined by (f◦g)(a)=f(g(a)) . (not defined unless the range of g is a subset of the domain of f)
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FUNCTIONS Examples: Let A = {1,2,3,4,5} B ={6,7,8,9} C= {10,11,12,13} D ={□,∆,◊,∂} Let R AxB, S BxC and T CxD be defined by R = {(1,7),(4,6),(5,6),(2,8)} S = {(6,10),(6,11),(7,10),(8,13)} T = {(11, ∆),(10, ∆),(13, ∂),(12, □),(13, ◊)} Compute the relations
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FUNCTIONS Definition The floor function that assigns to the real number x is the largest integer that is less than or equal to x . The value of the floor function at is denoted by Lx . The ˩ ceiling function assigns to the real number x the smallest integer that is greater than or equal to x . The value of the ceiling function at is denoted by Гx . ˥
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COMBINATORIAL STRUCTURES
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COUNTING Examples 1.
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