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| A number m in A is called a least element of A if x>m for every x in A. |
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| principle of mathematical induction |
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| For each positive integer n, let P(n) be a statement. If (1) P(1) is true and (2)the implication If P(k), then p(k+1). is true for every positive integer n. |
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| By a relation R from A to B we mean a subset of AxB. That is, R is a set of ordered pairs, where the first coordinate of the pair belongs to A and the second coordinate belongs to B. |
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is the subset of A defined by:
dom R={a in A:(a,b) in R for some b in B}; |
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| A relation R defined on a set A is called reflexive if x R x for every x in A. |
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| A relation R defined on a set A is called transitive if whenever xRy and yRz, then xRz, for all x,y,z in A. |
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| the set consists of all elements in A that are related to a. |
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| If m and n greater than or equal to 2 are integers and m is divided by n, then we can this division as m = nq + r, where q is the quotient and r is the remainder. The remainder r has to be 0 less than or equal to r |
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| If the sum(or product) of two equivalence classes does not depend on the representatives, then we say that this sum (or product) is well-defined. Ex. [a]=[b] and [c]=[d] in Z, then [a+c]=[b+d] and [ac]=[bd]. |
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| If (a,b) is in f, then we write b=f(a) and refer to b as the image of a. |
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| Define two functions to be equal, written f=g, if f(a)=g(a) for all a in A. |
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| If every two distinct elements od A have distinct elements w and z in A s.t. f(w)=f(z). |
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| if every element of the co domain B is the image of some element of A. |
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| For a nonempty set A, the function iA: A-A defined by iA(a)=(a) for each a is in A. |
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| Composition of the functions f,g,and h is associative if the functions h o (g o f) and (h o g) o f are equal. |
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| A permutation of a nonempty set is a bijective function on A, that is, a function from A to A that is both 1-1 and onto. |
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