Math / Discrete

Question $(A \cup(B \cap C))=$

Studdy Solution

STEP 1

What is this asking? We need to find which parts of the Venn diagram are included in $(A \cup (B \cap C))$ and add up the numbers in those parts. Watch out! Don't forget that the union symbol ( $\cup$ ) means "everything in either set" and the intersection symbol ( $\cap$ ) means "everything in both sets".
Don't mix them up!

STEP 2

1. Find the value of $B \cap C$ .
2. Find the value of $A \cup (B \cap C)$ .

STEP 3

Alright, so, $B \cap C$ means the parts where both $B$ and $C$ are present.
Looking at our awesome Venn diagram, that's the overlap between the $B$ and $C$ circles!

STEP 4

We see the number 9 in the part where only $B$ and $C$ overlap, and the number 1 where all three circles overlap.
Since both of these areas are in both $B$ and $C$ , we add them together: $9 + 1 = \mathbf{10}$ .
So, the value of $B \cap C$ is 10.

STEP 5

Now, we want $A \cup (B \cap C)$ .
This means everything in $A$ plus everything in $(B \cap C)$ , but without double-counting the overlap.
We already figured out that $(B \cap C)$ has a value of 10 (that's the overlap of $B$ and $C$ ).

STEP 6

$A$ includes the sections with 14 (only $A$ ), 8 (overlap of $A$ and $B$ ), 6 (overlap of $A$ and $C$ ), and 1 (overlap of all three).
Adding those up: $14 + 8 + 6 + 1 = \mathbf{29}$ .

STEP 7

Now, let's combine that with our value for $(B \cap C)$ , which is 10.
We have $A = 29$ and $(B \cap C) = 10$ .
We want $A \cup (B \cap C)$ .
Notice that the overlap of $A$ and $(B \cap C)$ is already included in our sum for $A$ (specifically, the 6 and the 1).
So, the parts of $(B \cap C)$ that aren't already counted in $A$ is just the part where only $B$ and $C$ overlap, which is 9.

STEP 8

Therefore, $A \cup (B \cap C)$ is $29$ (all of $A$ ) plus the part of $(B \cap C)$ that we haven't counted yet, which is $9$ .
So, $29 + (10 - (1+6)) = 29 + (10 - 7) = 29 + 3 = \mathbf{32}$ .

STEP 9

The value of $(A \cup (B \cap C))$ is 32.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms