Hello Sir,

Please explain the below formulae with their venn diagrams.

1) A-B=A⇔AintersectionB=Ø

2) Asubset of or equal toB⇔B'subset of or equal toA'

3) Aintersection(BincrementC)=(AintersectionB)increment(AintersectionC)

Asked by seeni2005 | 19th Aug, 2020, 09:19: PM

Expert Answer:

1) A-B=A⇔AintersectionB=Ø
 
Suppose space straight A minus straight B equals straight A rightwards double arrow No space elements space from space straight A space has space been space removed space when space we space remove space the space elements space of space straight B space from space straight A
This space means space that space there space is space no space element space in space straight B space which space is space common space to space straight A
rightwards double arrow There space is space no space element space common space in space straight A space and space straight B
rightwards double arrow straight A intersection straight B equals empty set

Let apostrophe straight s space assume space straight A intersection straight B equals empty set
rightwards double arrow space There space is space no space element space common space in space straight A space and space straight B comma space which space means space straight A space and space straight B space does space not space have space any space common space element
So comma space if space we space take space straight A minus straight B comma space this space means space we space are space deleting space the space elements space of space straight A space which space are space in space straight B space also.
But space as space there space is space no space element space of space straight A space which space is space in space straight B space rightwards double arrow space straight A minus straight B equals straight A space as space none space of space the space elements space of space straight A space has space been space deleted.
Hence comma space straight A minus straight B equals straight A left right double arrow straight A intersection straight B equals empty set
 

2) Asubset of or equal toB⇔B'subset of or equal toA'

Suppose space straight A subset of or equal to straight B rightwards double arrow space Either space straight A equals straight B space or space straight A subset of straight B
This space means comma space either space straight A space and space straight B space are space equal space or space straight B space contains space all space the space elements space of space straight A space and space straight A space does space not space have space all space the space elements space of space straight B
rightwards double arrow Either space straight A equals straight B space or space straight B superset of straight A
If space straight A equals straight B rightwards double arrow straight A apostrophe equals straight B apostrophe space... space left parenthesis straight i right parenthesis
straight A apostrophe space will space have space elements space not space in space straight A space and space straight B apostrophe space will space have space elements space not space in space straight B
As space straight A superset of straight B rightwards double arrow straight B apostrophe space will space have space elements space not space in space straight A space or space straight B
As space straight A subset of straight B rightwards double arrow straight A apostrophe space will space have space elements space not space in space straight A space but space will space have space some space elements space of space straight B space which space are space not space in space straight A.
rightwards double arrow straight A apostrophe superset of straight B apostrophe space... space left parenthesis ii right parenthesis
So comma space from space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space have
straight B apostrophe subset of or equal to straight A apostrophe

3) Aintersection(BincrementC)=(AintersectionB)increment(AintersectionC)

straight B triangle straight C equals open parentheses straight B minus straight C close parentheses union open parentheses straight C minus straight B close parentheses
straight A intersection open parentheses straight B triangle straight C close parentheses equals straight A intersection open square brackets open parentheses straight B minus straight C close parentheses union open parentheses straight C minus straight B close parentheses close square brackets
equals open square brackets straight A intersection open parentheses straight B minus straight C close parentheses close square brackets union open square brackets straight A intersection open parentheses straight C minus straight B close parentheses close square brackets

open parentheses straight A intersection straight B close parentheses triangle open parentheses straight A intersection straight C close parentheses equals open square brackets open parentheses straight A intersection straight B close parentheses minus open parentheses straight A intersection straight C close parentheses close square brackets union open square brackets open parentheses straight A intersection straight C close parentheses minus open parentheses straight A intersection straight B close parentheses close square brackets
equals open square brackets straight A intersection open parentheses straight B minus straight C close parentheses close square brackets union open square brackets straight A intersection open parentheses straight C minus straight B close parentheses close square brackets

Hence comma space straight A intersection open parentheses straight B triangle straight C close parentheses equals open parentheses straight A intersection straight B close parentheses triangle open parentheses straight A intersection straight C close parentheses

Answered by Renu Varma | 20th Aug, 2020, 11:38: AM