《离散数学》习题
第3章
1．Let Z be the set of all integers，and f：N?{0,1}，f(i)???1 i为奇数 ?0 i为偶数，then f（ ）．
(A)．is a injection and not a surjection； (B)．is a surjection and not a injection；
(C)．is neither a injection nor a surjection； (D)．is a bijection．
2．Let R be the set of all real numbers，andf：R?R，f(x)?2x，thenf（ ）．
(A)．is a injection and not a surjection； (B)．is a surjection and not a injection；
(C)．is neither a injection nor a surjection； (D)．is a bijection．
3．Letf,g,h are relations on setA． Which of the following propositions is ture?（ ）．
(A)． f?g?g?f ； (B)．f?f?f； (C)．f?(g?h)?(f?g)?h； (D)．f?g?h．
4．If f be a bijection fromA toB，then ff?1?1 is a bijection from ，and
?f= ，f?f?1= ．
5．Let A?{a,b,c}，then the number of bijections from A to A is ．
6．Let X?{a,b,c,d}, Y?{1,2,3,4}． Which of the following relations from X to Y are or are not functions? Find Rf for functions．
f1={a,1,b,3,c,4,d,2f2},
={a,4,b,4,c,4,d,4},
},
f3={a,3,b,4,c,3,d,4f4={a,4,b,4,c,1,d,4},
}
f5={a,1,b,3,c,4,d,2,a,2f6={a,1,b,3,c,4}
7．Let A?{a,b,c},B?{1,2},C?{?,?}，
f:A?B,f?{a,1,b,2,c,2}，g:B?C,g?{1,?,2,?}，
find g?f:A?C．
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8．Let R? be the set of all positive real numbers and R be the set of all real numbers，f：
R???R,?x?R,f(x)?lnx． Show that f is a bijection．
9．Let Rbe the set of all real numbers，a,b?R,a?b,f：[a,b]?[0,1]， f(x)??x?[a,b]．
x?ab?a，
Show thatf is a bijection and find f?1．
第4章
1．Let G,? is a group with identity 1，where G?{1,a,b}，then b2= , a3= ．
2．Suppose G,? is a group，then for all a,b∈G, (a?b)?1 = ．
3．In algebraic systemZ,?（Zis the set of all integers and “+” is the general addition），the identity for Z with respect to the operation “+” is ，the zero element for Zwith respect to the operation “+” is ，for any x∈Z，x?1＝ ． 4．Let A?{a,b,c}and for
P(A)
P(A)
be the power set of A．In algebraic system?P(A),?,??，the identity
P(A)
with respect to the operation “?” is ，the zero element for
P(A)
with respect to
the operation “?” is ，the identity for zero element for
P(A)
with respect to the operation “?” is ，the
with respect to the operation “?” is ．
5．Which of the following algebraic systems ?G,?? is not a group（ ）．
(A)．G＝{1,10}，? is defined by m?n=mn (mod 11)；(B)．G={1,3,4,5,9}, ? is the same as A； (C)．G?Q，? is the general multiplication； (D)．G?Q，? is the general addition． 6．Let ? be a binary algebraic operation(二元代数运算) on setA． ??A，? is called a zero element forAunder “?” if（ ）．
(A)．for all x?A，such that ??x?x???x． (B)．for all x?A，such that ??x?x????．
(C)．there is an element x?A，such that ??x?x???x． (D)．there is an element x?A，such that ??x?x????． 7．Let ? be a binary algebraic operation(二元代数运算) on setA． e?A，e is called an identity (element) forAunder “?” if（ ）．
(A)．for all x?A，such that e?x?x?e?x． (B)．for all x?A，such that e?x?x?e?e．
(C)．there is an element x?A，such that e?x?x?e?x． (D)．there is an element x?A，such that e?x?x?e?e．
8．Let Q be the set of all rational numbers． We define operation “?” by
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?a,b?Q,a?b?a?b?2ab ．
(1)．Find 3?(?5)；
（2）Find the identity for Q with respect to the operation ?； (3)．Does ain Q have an inverse? If yes , please find a?1．
9．Fill the following blanks with proper elements so that {a,b,c},* becomes a group．
? a b c a ab a c c c
10．LetA?{a,b,c,d}， A,? be an Abelian group and abe the identity of A,?．The binary operation “?” defined by
? a b c d aa b c d b b a x1 x2cc x4 a x3 d x5 x6 ad Find x1,x2,x3,x4,x5,x6．
11．Let G,? is a monoid with identity e ，and ?a?G，a?a?e，then G,? is an Abelian group．
12．Let S?{a,b,c,d}, f:S?S be a bijection , f(a)?b,f(b)?c,f(c)?d,f(d)?a，and
F?{f,f,f,f}．
0123Show thatF,?is an Abelian group．
第5章
1．〈A,≤〉be a poset，if any two elements in A have ，then
〈A,≤〉is called a lattice， and ?a,b?A，a?b if and only if a?b? ．
2．Let A,?,?,,0,1 be a Boolean algebra，then there are deferent functions from
nA to A； the identity for A with respect to the operation “?” is ，the zero element
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for A with respect to the operation “?” is ，the identity for A with respect to the operation “?” is ，the zero element for A with respect to the operation “?” is ．
3．Let A,?,?,,0,1 be a Boolean algebra．For all a,b?A，(a?b)?____ __． 4．Which of the following Hasse diagrams is not a lattice? （ ）．
5．Which of the following poset is bounded lattice? （ ）
(A)．?N,??； (B)．??2,3,4,6,12?,／?，where “／” is the relation of exact division； (C)．?Z,??； (D)．?P(A),??；where A?{a,b,c}，P(A) is the power set of A． 6．Which of the following Hasse diagrams is not a distributive lattice? （ ）．
7．Which of the following Hasse diagrams is a distributive lattice?（ ）
8．Which of the following Hasse diagrams is a distributive lattice? （ ）．
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9．Let L,?be a lattice defined by the following Diagram． Find complements (if there exist) for
every element of L．
Solution： element complements 0 1 a b c d e 10．Let L,?be a lattice defined by the following Diagram． Find complements (if there exist) for every element of L．
Solution： element a b c d e （1分） c,d（2分） b（1分） b（1分） a（1分） complements e