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‘fromfirst principles’that A=B.

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

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

(b)Prove‘fromfirst 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}iffa=b.

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