Miscellaneous Facts about Functions
Miscellaneous Facts about Functions
复制标题
关于函数的其他事实
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
A. Trybulec
中科院分区：
文献类型：
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作者：
G. Bancerek;A. Trybulec
为简单起见，我们采用以下规则：X是任意的，m，n是自然数，f，g是函数，A，B是集合。我们现在陈述几个命题：(1)对于每个函数f和每个集合X，使得RNG f⊆X满足idx·f=f。(2)设X是集合，Y是非空集，f是从X到Y的函数。假设f是一对一。设B是X的子集，C是Y的子集。如果C⊆f B，则f−1 C⊆B。(3)设X，Y是非空集，f是从X到Y的函数。假设f是一对一。设x是X的一个元素，A是X的一个子集，如果f(X)∈f A，则x∈A。(4)设X，Y是非空集，f是从X到Y的函数。假设f是一对一。设X是X的一个元素，A是X的子集，B是Y的子集。如果f(X)∈f A\B，则x∈A\f−1 B。(5)设X，Y是非空集，f是从X到Y的函数。假设f是一对一。设y是Y的一个元素，A是X的一个子集，B是Y的一个子集。如果y∈f A\B，则f−1(Y)∈A\f−1B.(6)对于每个函数f和任意的a，使得∈Dom f满足f{a}=A7−→。F(A)。
For simplicity we adopt the following rules: x is arbitrary, m, n are natural numbers, f , g are functions, and A, B are sets. We now state several propositions: (1) For every function f and for every set X such that rng f ⊆ X holds idX · f = f. (2) Let X be a set, and let Y be a non empty set, and let f be a function from X into Y . Suppose f is one-to-one. Let B be a subset of X and let C be a subset of Y . If C ⊆ f B, then f −1 C ⊆ B. (3) Let X, Y be non empty sets and let f be a function from X into Y . Suppose f is one-to-one. Let x be an element of X and let A be a subset of X. If f(x) ∈ f A, then x ∈ A. (4) Let X, Y be non empty sets and let f be a function from X into Y . Suppose f is one-to-one. Let x be an element of X, and let A be a subset of X, and let B be a subset of Y . If f(x) ∈ f A \B, then x ∈ A \ f −1 B. (5) Let X, Y be non empty sets and let f be a function from X into Y . Suppose f is one-to-one. Let y be an element of Y , and let A be a subset of X, and let B be a subset of Y . If y ∈ f A\B, then f−1(y) ∈ A\f −1B. (6) For every function f and for arbitrary a such that a ∈ dom f holds f {a} = a7−→ . f(a).