1968-12-19 — Page 27

{# (ANB)+(ANG)-

Intersection is distributive

over difference

...Def.of symmetrical

differença

Prove that A-(BNC) = (A−B]U{A=C),

Proof :

Left hand member »AN(BNC)

= (AMB' JU(ANC)) (A–B)U (A=§)

..Def of difference

De Morgan's law

Distributive law-

...Def.of difference

Prove that (A-B-C Proof:

Left hand member

(AMB)NC

(ANC)AB

(ANG)-B

(d) Prove that A-(B-A)

Proof:

Left hand pa

∙AN(BUK) AKAUBU)

(A/C)-B

„Def.of difference.

Associative law

Def. of difference

„Def.of difference »«De Morgan's law „Commutative law Absorption law.

*) Prove that (ANB)U(B-A) = (AUB)

Proofs

Left hand member

= (AUB)U(BŊA').

Def of difference

[(AUB)UB]N[(AUB]JA*]

...Distributive law

(AUB)n [(AUA ® )UB]..Idempotent law.

vi Commutativek ...associative law

(AUB){[(UVB)})}

CAUB)(T

AUB

...Comptement law

...Identity law

Identity law

()Prove that (A−B){}{G+" ) = (ATC)~(BID}

Proof:

Left hand member

{ÂNB'}/{CAD)

CANG )N(B*AD®)

= (ANC)A(BUD)"

= (ANC)=(BUD)

Def.of differenci Commutative &

Associative lau

De Morgan's law

Def. of difference

Prove that A-[B={C_D}}= (A−B}U[(ANG)=D}

Proof:

Left hand member

= A^[B^{CAD®)']'

AN[B/KCUD)] •

= AN[BUCCOD

* {AÑB®)UCANCÍD

=(A-B)U[(ANC)-D]

(b) Simplify each of the

•A+Ø = {A=Ø)U(Ø=A),

(ANS) JULØNA) (ANU) URBAN).

AUA

A+A® = {A¬A ̈}U¢Ã÷ATM)

...Def.or differenci

„De morgan's law

De Morgan's law. ...Distributive lay

.Def of difference

following:

Def of symmetrical

differance

Def.of difference „Complement law

Identity law Complement law

sf of Bymmetrical

difference

{AÑA JUCAÑA!) :Def.of differenc

A+U = (A=0){{U-A)

(AND YUTURA"), KANDUCATOR

{ A÷A® ) U¢Â ́~A

(AAA)UCA VIA®

Power Set of a Set

Idempotent lav

■Complement lav

Def.of symmetrical

difference Def of difference Complement &

dentity laws Identity laws Identity law

..Def.of symmetrical

Difference

Def.of difference Idempotent law Complement law

Definition: Given a set the subsets of T, denoted the power set of T.

set of all is called

Notice that 2

is a symbol for set it is not a power of 2 Examples

(1) Find the power set of A, where A

Solution:

and:

The subsets of A are {],108, 10.18.

Therefore the power set of

(2) Find the power set of B where B={1,1,1}

Solution:

The subsets of B are {},{1},{0},{2}, {1,m},{«,n},{\‚n}, and {1,v,n) - Hence,

Find the number of elements of the power set of C, where Ca{1,2,3,4] Solution:

The number of elements in 2 can be found by branching sketches as follows.

Let "7" denotes the element is taken and "10" denotes the element is not

Isete

- {1,2,3,4) ..{1,2,3}

{1,2}

{+3,4}

(4)

·(2,3,4) (2.3) (2.4)

{2}

Therefore, there are 16 elementa in

{3} (4)

În general, if the number of elements in a iven set S is n, then 2a has 2′′ elements.

Cartesian set of two seta

Definition: The Cartesian set AzB of two sets A and B is the set of all ordered pairs (x,y), such, that x belongs to. A and y belongs to B. That is

AxB = {(x,y}} XEA and yEB}

Examples:

Find the Cartesian set CxD and D= {1,b,c}

Solution:

CZD ={(x,y}{xƐC and yED}

四期星

日九十月二十年八六九一万个年七十五家民营中

三九六九市中玄中学金者試題預

英文科

(E)

王淑方

LESSON SEVEN.

CHAPTER TWO

16-12-1968,

Direct Speech

C= {1,2}

(B)

•{(1,a),(1,5),(1,c),(2,a),(2,b),(2,0){

(2) Find the Cartesian set MXM, where

M = {1,2,5,6}

Solution:

Mel (x,y]]xEM and YEN

11,1),(1,2),(1,5),(1,6),(2,1),(2,2),

(2,5),(2,6), (5,1),(5,2), (5.5), (5,6).

· (6,1),(6,2),(6,5),(6,6)}

(3) Frove that Ax(BNC) = (AxB)(AxC)

To prove the statement le to provai

(a) Ax(B/C)C(AxB}/}(AxC)

(b) (ARB)/(\xC) = Ax (BIG)

Proof:

and

(a) Ley (x,y) be any element in Ax(BAC),

Then,

XEA and yE(BAC)

That means

YEB and yếu

Hence

(j){AB}

and (x,y)E(AC)

Therefore,

..Def. of

Cartesian set

Def.of intersection.

In direct speech we have the exact word the speaker.

"I am writing an essay," said Anni,

Indirect Speech

In Indirert Speech (Reported Speech) ve may report what the speaker said without quoting his exact words; we give the same meaning but with a different form. Thus, the words spoken are in- corporated into the structure of the main sentence. Ann said that she was writing an essay.

The difference between the two forms is shown

jec

by the tense of the verb, together with changes, in

the person of the pronouns, and possessi aves and of certain words that denoted in the direct form. There are also in

tungek in word order MORAT RULES FOR CHANGING DIRECT SPEECH INTO. INDIRECT

Def.of Cartesian

NOTE:

Ax(BIC) (AxB){\(AxC) ........Def.of

Ben Cartesian set

(b) Let (x,y)E(AxB)/(4xC). Then,

(x,y)E(AxB) and

*,y)E(AxC)

That 16, XE A and yeB

and also yEC

Henee

ye(BAC)

Therefore,

(x,y)£\x{B/C)

Hence,

(AxB)(AXC)C AX(B/C)

Def of intersection

Def of Cartesian

·set

Def.of intersectiu

Changes in ferosi

Simple Present Tense becomes Simple Past Tense.

(DIREXT)

"I write a letter every week," she

said.

(AND IRECT) |

She said that she wrote a Jefter

every week..

11): Present Continuous becomes Past Continuous

(DIRECT);

"I am reading a book, he said. (INDIRECT):

He said that he was reading a boc

(111) Present Perfect becomes Past Perfec

(DIRECT)

"I have read a book,“ she said INDIRECT) 1

She said that she had read i

Past Tense becomes Fist Perfact

(DIRECT Y

"I wrote a letter

(INDIRECT):

He said that he had written

ire Tense becomes Future in the "Fast

Lie, shally

should would

"I shall see the beachsagter – he said. (DIRECT

He said that he would see the head-

master. I TADTREET)

Conditional becomes Perfect Conditional

(DIRECT

If I had my

the story

INDIRECT),

I could read

He said that if he had had his book, he could have read the story.

Changes in: Pronouns and Possessive Adjectives

Pronouns and Possessive Adjectives; of the

and Second Persons, are changed to the

DIRECT

Ours

you your

your

6.g.

(DIRECT SPEECH)

ne (she)

bra (her

they

their theirs

they then their

theire

(a) "I bring my pens every day," he said. (b) The pen on the desk, la mine, he said. {c} "We bring our pens every day, they saidi (d) "The pens on the deak are ours They

said.

(INDIRECT SPEECH).

(a) He said that

day

broug his pena every:

(b) He said that the pen on the desk was hist (c) They said that they brought their pens.

every day.

(d) They said that, the pers on the desk were

theirs.

Some of these pronouns and possessive adjectiven may vary according to circumstance. Commor sense will determine which pronouns should used.

- you ? may be changed into he

*your* "may be changed into-

Def.of Cartesian

Det

(ii) you

Def.of

bae*

Exercise for the week

for

(1) Given: A={1,2}, B={b,c,d}, C=ja,0,0}, find AxBx0.

and

(2) What is the number of elements in XxX 11 X. has 4 elements and Y has 3 elementa? Why?

(3) Given A, and B={1,B,nf, find AxB. (4) Given Sc.(0,1,2,4), find

(5) Given A={0}, find 2

(6) Given K={a,b,c,d}, indicate which of

the following are trues KE2K; {S}c2 bɛ 2K

(DIRECT SPEECH)}}T

The teacher told Jack, "You must bring vour note-book to school.

INDIRECT SPEECH)

The teacher told Jack that he must bring bes note book to school.

* may be changed into this,, 'here' or

DIRECT SPEECH)•

Jack asked Jane, "Is this book yours? AINDIRECT SPEECH I

Jack asked Jane if that book was horas

ANSWERS TO EXERCISES EXERCISE 8 (1) (A) (2) (D) EXERCISE 19 (1) (B) M (2) (A) (3)(B)

(4)

1(4) (A)

(5) (D)

(6) (0) (7) (R) (8) (P)) (9) (T)| (10WS)!

88.0

(11)(B) (12) (5) (13)X(E) (14)4(0) (15) (A)

(P):

Comments

Approved members can add comments, bookmarks, and private notes.

No comments yet.

Private Research Note

Private notes are available after approval.