香港中文大学数学课程-现代数学奠基(MATH1050) 额外练习 (二)

MATH1050B/C Further Exercise2Due date:20-2-2017

1.Let P,Q,R be statements.Consider each of the pairs of statements below.Determine whether the statements are

logically equivalent.Justify your answer by drawing an appropriate truth table.

(a)～(P∨Q),(～P)∧(～Q).

(b)P→(Q∧R),(P→Q)∧(P→R).

(c)P→(Q→R),(P∧Q)→R.

(d)P→(Q∨R),(P→Q)∨(P→R).

(e)(P∨Q)→R,(P→R)∧(Q→R).

(f)(P→Q)→R,P→(Q→R).

(g)P→(Q∨R),[P∧(～Q)]→R.

2.Let P,Q,R be statements.Consider each of the statements below.Determine whether it is a tautology or a

contradiction or a contingent statement.Justify your answer by drawing an appropriate truth table.

(a)[P→(P→Q)]→(P→Q)

(b)(P→R)→[(P∧Q)→R)]

(c)[(P→Q)∧(Q→R)]→(P→R)

(d)[(P→Q)∧(Q→R)∧(R→P)]→(Q→P)

(e)(P→R)→[(P→Q)∨(Q→R)]

(f)(P→Q)→[(Q→R)∨(P∧R)]

(g)(P→Q)→[(P→R)∨(Q→R)]

3.Let C={0,1,1,2,3,3,4},D={0,1,{1,2,3},{{3},4}}.Consider each of the sets below.List every element of

the set concerned,each element exactly once.You are not required to justify your answer.

(a)C.

(b)D.(c)C∩D.

(d)C∪D.

(e)C\D.

(f)D\C.

(g)C△D.

(h)P(C∩D).

4.Let C={{0,1},{1},{1,2,3},{3,4}},D={{0,1,1},{1,2,3},{{3},{4}}}.

Consider each of the sets below.List every element of the set concerned,each element exactly once.You are not required to justify your answer.

(a)C∩D.(b)C∪D.(c)C\D.(d)C△D.(e)P(C\D).

5.Let M={m,a,r,c,u,s},T={t,u,l,l,i,u,s},C={c,i,c,e,r,o}.

(a)How many elements are there in the set C?

(b)How many elements are there in the set M∪T?

(c)How many elements are there in the set(M∪T)\C?

(d)How many elements are there in the set{(M∪T)\C}?

(e)How many elements are there in the set({M}∪{T})\{C}?

(f)How many elements are there in the set{M∪T}\{C}?

(g)List every element of the set M∩C,each element exactly once.

(h)List every element of the set P(M∩C),each element exactly once.

6.Let A={x∈R:x2?2x?3≤0},B={x∈R:?1≤x≤3}.Prove‘from?rst principles’that A=B.

7.Let A={n∈Z:n≡1(mod3)},B={n∈Z:n≡4(mod9)}.

(a)Prove‘from?rst principles’that B?A.

(b)?Is it true that A?/B?Justify your answer.

8.Let A={x∈Z:x=k4for some k∈Z},B={x∈Z:x=k2for some k∈Z}.

(a)Prove that A?B.

(b)?Is it true that B?/A?Justify your answer.

9.?Let A={x∈R:x2?x≥0},B={x∈R:x≤0},C={x∈R:x≥1}.Prove‘from?rst principles’that

A=B∪C.

10.Let A={x∈Q:x=r3for some r∈Q},B={x∈Q:x=r9for some r∈Q},C={x∈Z:x=

r3for some r∈Q},D={x∈Q:x=r3for some r∈Z}.

(a)Is A a subset of Q?Is Q a subset of A?Justify your answer.

(b)Is A a subset of B?Is B a subset of A?Justify your answer.

(c)Is A a subset of C?Is C a subset of A?Justify your answer.

(d)Is A a subset of D?Is D a subset of A?Justify your answer.

11.To handle this question,you may make use of what you have learnt in‘linear algebra’and/or‘coordinate geometry’

and/or‘vector geometry’.

(a)Let A={p∈R3:There exist some x,y,z∈R such that p=(x,y,z)and4x+2y+z=0},B={q∈R3:

There exist some s,t∈R such that q=(s,t,?4s?2t)}.

Prove‘from?rst principles’that A=B.

Remark.What are A,B really?

(b)Let A={p∈R3:There exist some x,y,z∈R such that p=(x,y,z)and2x+y?z=0and x?2y+z=0},

B={q∈R3:There exists some t∈R such that q=(t,3t,5t)}.

Prove‘from?rst principles’that A=B.

Remark.What are A,B really?

12.?To handle this question,you may make use of what you have learnt in‘linear algebra’and/or‘coordinate geometry’

and/or‘vector geometry’.

Let

S={p∈R3:There exist some x,y,z∈R such that p=(x,y,z)and x2+y2+z2=1},

H={p∈R3:There exist some x,y,z∈R such that p=(x,y,z)and x2+y2?z2=1},

C={p∈R3:There exist some x,y,z∈R such that p=(x,y,z)and x2+y2=1},

Γ={q∈R3:There exist someθ∈R such that q=(cos(θ),sin(θ),0)}.

(a)Prove‘from?rst principles’thatΓis a subset of each of S,H,C.

(b)Prove‘from?rst principles’that each of S∩H,H∩C,C∩S is a subset ofΓ.

(c)Deduce thatΓ=S∩H=H∩C=C∩S.

Remark.What are S,H,C,Γreally?Make use of what you have learnt in coordinate geometry.

13.?In this question,you may use of the following statements without proof:

(?)Let a,b be two objects(not necessarily distinct).{a}={b}i?a=b.

(?)Let a,b,c be three objects(not necessarily distinct).{a,b}={c}i?a=b=c.

(?)?={?}.

Let A={?},B={{?}},C={?,{?,{?}}},D={?,{{?}}}.For each of the following statements,determine whether it is true or false.Prove your answer in each case.

(a)A∈C.

(b)B?D.(c)B∈C.

(d)A∪B∈C.

(e)C∩D=?.

14.Prove each of the statements below‘from?rst principles’,using the de?nitions of set equality,subset relation,

intersection,union,complement,where appropriate.

(a)Let A,B be sets.A∩B?A.

(b)Let A,B be sets.A?A∪B.

(c)Let A,B,C be sets.A\(B∪C)=(A\B)∩(A\C).

(d)Let A,B,C be sets.(A∪B)\C=(A\C)∪(B\C).

(e)Let A,B be sets.(A∪B)\A=B\(A∩B).

(f)Let A,B,C be sets.A\(B\C)=(A\B)∪(A∩C).

15.Prove the following statements:

(a)?Let A,B be sets.A△B=(A∪B)\(A∩B).

(b)Let A,B be sets.A△B=B△A.

(c)?Let A,B,C be sets.(A△B)△C=A△(B△C).

(d)?Let A,B,C be sets.A∩(B△C)=(A∩B)△(A∩C).

16.?Let A,B be sets.Prove that the following statements are equivalent:

(I)A?B.

(II)A∩B=A.

(III)A∪B=B.

17.Dis-prove each of the statements below by giving an appropriate counter-example.(It may help if you draw Venn

diagrams to investigate the respective statements?rst.)

(a)Let A,B,C be sets.A\(C\B)?A∩B.

(b)Let A,B,C be non-empty sets.B\A?(C\A)\(C\B).

(c)Let A,B,C be non-empty sets.A∪(B∩C)?(A∪B)∩C.

(d)?Let A,B,C are non-empty sets.B∩C?[A\(B\C)]∪[B\(C\A)].

(e)?Let A,B,C be sets.Suppose A∩B?C.Then C?(A∩C)∪(B∩C).

18.?Consider each of the statements below.In each case,determine whether it is true or false.Justify your answer

by giving a proof or constructing a counter-example where appropriate.

(a)Let A,B,C be sets.Suppose A∪(B∩C)=(A∪B)∩C.Then A?C.

(b)Let A,B,C be sets.A\(B\C)=(A\B)\C.

(c)Let A,B,C be sets.If A?B then C\A?C\B.

(d)Let A,B,C be sets.Suppose A?B and A?/C.Then B?/C.

(e)Let A,B,C be non-empty sets.Suppose A?B and B?/C.Then A?/C.

(f)Let A,B,C be non-empty sets.Suppose A?B and B?/C.Then A?/C.

(g)Let A,B,C be sets.Then A∪(B△C)=(A△B)∪(A△C).

(h)Let A,B,C be sets.Then A∩(B△C)=(A△B)∩(A△C).

19.(a)Prove the statements below‘from?rst principles’,using the de?nitions of set equality,subset relation,inter-

section,union,complement,where appropriate.

i.Let E be a set,and A,B be subsets of E.Suppose A?B.Then E\B?E\A.

ii.Let E be a set,and A,B be subsets of E.Suppose A B.Then E\B E\A.

(b)Consider each of the statements below.For each of them,determine whether it is true or false.Justify your

answer by giving a proof or constructing a counter-example where appropriate.

i.Let A,B,E be a set.Suppose A?B.Then E\B?E\A.

ii.Let A,B,E be a set.Suppose A B.Then E\B E\A.

20.(a)?Consider each of the statements below.For each of them,determine whether it is true or false.Justify your

answer by giving a proof or constructing a counter-example where appropriate.

i.Let A,B be sets.B\(B\A)?A.

ii.Let A,B be sets.A?B\(B\A).

(b)?Prove the statements below:

i.Let A,B be sets.A?B\(B\A)i?A?B.

ii.Let A,B be sets.B\(B\A)=A i?A?B.

iii.Let A,B be sets.B\(B\A) A i?A?/B.

21.Prove the following statements:

(a)?Let A,B be sets.P(A\B)?(P(A)\P(B))∪{?}.

(b)?Let A,B be sets.Suppose(A?B or A∩B=?).Then P(A)\P(B)?P(A\B).

(c)?Let A,B be sets.Suppose P(A)\P(B)?P(A\B).Then(A?B or A∩B=?).

22.Prove the following statements:

(a)Let a,b be two objects(not necessarily distinct).{a}={b}i?a=b.

(b)Let a,b,c be three objects(not necessarily distinct).{a,b}={c}i?a=b=c.

(c)?Let a,b,c,d be four objects(not necessarily distinct).{a,b}={c,d}i?((a=c and b=d)or(a=d and

b=c)or a=b=c=d).

(d)?Let A,B,C,D be sets.Suppose{A,B}={C,D}.Then A∩B=C∩D and A∪B=C∪D.

23.Consider the predicate‘x/∈x’,which we denote by P(x).

(a)Denote by R the object{x|P(x)},obtained from the Method of Speci?cation.(R is called the Russell set.)

Suppose it were true that R was a set.(Hence it makes sense to discuss whether an arbitrarily given object is an element of R or not.)

i.Can it happen that the object R is an element of the set R?Why?

ii.Can it happen that the object R is an element of the set R?Why?

Remark.From the answers to the above questions,you would have to conclude that R is not a set in the ?rst place.(Why?)This tells us the construction{x|P(x)}fails to give a set.

(b)Let A be a set.Denote by B the object{x∈A:P(x)},obtained from the Method of Speci?cation.(This

time it is guaranteed that B is a set,because we are constructing a subset from the given set A.)

Prove that B is not an element of A.(Apply the proof-by-contradiction method.)

Remark.This shows that given any set,there is always some object which does not belong to it as an element.In other words,no set contains every conceivable object as its element.There is no such thing as ‘universal set’.