Question

please get me started :

Find a set expression that corresponds to the shaded region shown in the Venn diagram below.

Answer 1

there are probably infinitely many ways to do this, but one way is to union those 3 regions

Answer 2

so you need to find expressions for each region

Answer 3

ok and how do i start that

Answer 4

the very middle piece (the smallest shaded region) is the result of intersecting the 3 circles

Answer 5

basically, it's the region common to all 3 circles

Answer 6

right

Answer 7

so that region is

\[\Large (A \cap B) \cap C\]

Answer 8

ie, A intersect B intersect C

Answer 9

now you need to find the other two shaded regions, they are very similar, so that's a bit of good news

Answer 10

ok i see that

Answer 11

look at the upper shaded region

Answer 12

b?

Answer 13

yes for set B

Answer 14

ok

Answer 15

notice how everything is filled in but stuff from sets A or C (ignore the common middle right now)

Answer 16

yes

Answer 17

so that region is basically: everything in set B but kick out stuff from set A or C

which turns into

\[\Large B \cap (A \cup C)'\]

Answer 18

ok

Answer 19

you are basically intersecting set B with everything that is in neither A nor C, that gives you that upper shaded region in B

Answer 20

ok

Answer 21

do the same for that last shaded region

Answer 22

A∩(B∪C)′

Answer 23

good

Answer 24

now we have the following regions:

---------------------------------------------------

\[\Large (A \cap B) \cap C\]

which is the smallest shaded region (in the middle)

---------------------------------------------------

\[\Large B \cap (A \cup C)'\]

the upper shaded region found in set B

---------------------------------------------------

\[\Large A \cap (B \cup C)'\]

the lower shaded region found in set A

Answer 25

right

Answer 26

we then union those regions together to get the final answer

Answer 27

ok and what does that mean

Answer 28

let me show you, one sec

Answer 29

ok thanks

Answer 30

if you color and label the regions like this

Answer 31

ok

Answer 32

and then just label the equations ewith the regions?

Answer 33

or do we combine them, ?

Answer 34

then you can say this:

Answer 35

and thats the answer ?

Answer 36

then union them to get this

Answer 37

so your final answer is this

Answer 38

ok great! thanks so much!!

Answer 39

me personally, i like the labels and the colors (so you know what's what), but you'll probably have to drop them

Answer 40

ok great thanks so much!