Venn Diagram Worksheets | Dynamically Created Venn Diagram Worksheets - Free Printable

Let's solve each problem step by step using the Venn diagrams provided.

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Problem 1

Venn Diagram: Sets K and L

- Elements in K only: 20, 16, 17
- Elements in L only: 5, 11, 1
- Elements in both K and L (intersection): 4, 3, 14
- Outside both sets: 6, 10, 18

We are asked to find:

#### 1) $ K' \cap L' $

- $ K' $: elements not in K
- $ L' $: elements not in L
- $ K' \cap L' $: elements not in K or L, i.e., outside both circles

From diagram: 6, 10, 18

✔ So, $ K' \cap L' = \{6, 10, 18\} $

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#### 2) $ L - K' $

This means: elements in L but not in K'

But note:
- $ K' $ is everything not in K
- So $ L - K' $ = elements in L that are in K (because if they're not in $ K' $, they must be in K)

So: $ L - K' = L \cap K $

Which is the intersection: 4, 3, 14

✔ $ L - K' = \{4, 3, 14\} $

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#### 3) $ L - K $

Elements in L but not in K

That’s the part of L outside K → 5, 11, 1

✔ $ L - K = \{5, 11, 1\} $

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#### 4) $ K $

All elements in set K

K includes: 20, 16, 17, 4, 3, 14

✔ $ K = \{20, 16, 17, 4, 3, 14\} $

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✔ Answers for Problem 1:

- $ K' \cap L' = \{6, 10, 18\} $
- $ L - K' = \{4, 3, 14\} $
- $ L - K = \{5, 11, 1\} $
- $ K = \{20, 16, 17, 4, 3, 14\} $

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Problem 2

Venn Diagram: Sets G and H

Let’s label regions:

- Only G: n, x, q
- Only H: r, b, p
- Intersection: g, h, z
- Outside both: c, e, a

Now solve:

#### 1) $ G \cap H $

Intersection of G and H → elements in both sets

→ g, h, z

✔ $ G \cap H = \{g, h, z\} $

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#### 2) $ G' \cap H' $

Complement of G and complement of H → elements not in G and not in H

Outside both circles: c, e, a

✔ $ G' \cap H' = \{c, e, a\} $

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#### 3) $ H' - G' $

$ H' $: not in H → elements outside H

→ n, x, q, c, e, a

$ G' $: not in G → r, b, p, c, e, a

Now $ H' - G' $: elements in $ H' $ but not in $ G' $

So remove from $ H' $ those that are in $ G' $

$ H' = \{n, x, q, c, e, a\} $

$ G' = \{r, b, p, c, e, a\} $

Common: c, e, a → remove them

Left: n, x, q

These are elements in H' but not in G' → which means they are in G (since not in G') and not in H → so they are only in G

Yes! So $ H' - G' = \{n, x, q\} $

✔ $ H' - G' = \{n, x, q\} $

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#### 4) $ (G - H)' $

First: $ G - H $ = elements in G but not in H

→ n, x, q

So $ G - H = \{n, x, q\} $

Now take complement: $ (G - H)' $ = everything not in $ \{n, x, q\} $

So all other elements: g, h, z, r, b, p, c, e, a

Note: This includes everything except n, x, q

✔ $ (G - H)' = \{g, h, z, r, b, p, c, e, a\} $

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✔ Answers for Problem 2:

- $ G \cap H = \{g, h, z\} $
- $ G' \cap H' = \{c, e, a\} $
- $ H' - G' = \{n, x, q\} $
- $ (G - H)' = \{g, h, z, r, b, p, c, e, a\} $

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Problem 3

Venn Diagram: Sets A and B

Numbers given:

- Only A: 95, 20, 55
- Only B: 85, 40
- Intersection: 75, 70, 80
- Outside both: 25, 50, 15

Total universe: sum of all numbers = ?

But we don't need total unless asked.

Now solve:

#### 1) $ A \cup B $

Union of A and B → all elements in A or B or both

= All inside either circle

= 95 + 20 + 55 + 75 + 70 + 80 + 85 + 40

But let’s list them clearly:

- A only: 95, 20, 55
- B only: 85, 40
- Both: 75, 70, 80

So $ A \cup B = \{95, 20, 55, 75, 70, 80, 85, 40\} $

But since these are numbers, we can write as set of values.

✔ $ A \cup B = \{20, 40, 55, 70, 75, 80, 85, 95\} $

(Or just list all values; order doesn’t matter)

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#### 2) $ B' $

Complement of B → everything not in B

B includes: 75, 70, 80, 85, 40

So $ B' $ = all not in B

→ 95, 20, 55, 25, 50, 15

Also check: outside both and only A

So $ B' = \{95, 20, 55, 25, 50, 15\} $

✔ $ B' = \{15, 20, 25, 50, 55, 95\} $

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#### 3) $ B - A $

Elements in B but not in A

→ B only: 85, 40

Because intersection is in both, so not included

✔ $ B - A = \{40, 85\} $

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#### 4) $ A' \cup B' $

Use De Morgan’s Law: $ A' \cup B' = (A \cap B)' $

But let's compute directly.

$ A' $: not in A → elements not in A

→ 85, 40, 25, 50, 15

$ B' $: not in B → 95, 20, 55, 25, 50, 15

Now $ A' \cup B' $: all elements in either $ A' $ or $ B' $

Combine: 85, 40, 25, 50, 15, 95, 20, 55

→ Remove duplicates: 15, 20, 25, 40, 50, 55, 85, 95

So $ A' \cup B' = \{15, 20, 25, 40, 50, 55, 85, 95\} $

Alternatively, this is everything not in A ∩ B

$ A \cap B = \{75, 70, 80\} $

So $ (A \cap B)' $ = all except 75, 70, 80

→ All others: 95, 20, 55, 25, 50, 15, 85, 40

Same result.

✔ $ A' \cup B' = \{15, 20, 25, 40, 50, 55, 85, 95\} $

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✔ Answers for Problem 3:

- $ A \cup B = \{20, 40, 55, 70, 75, 80, 85, 95\} $
- $ B' = \{15, 20, 25, 50, 55, 95\} $
- $ B - A = \{40, 85\} $
- $ A' \cup B' = \{15, 20, 25, 40, 50, 55, 85, 95\} $

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✔ Final Answer Summary:

#### Problem 1
- $ K' \cap L' = \{6, 10, 18\} $
- $ L - K' = \{4, 3, 14\} $
- $ L - K = \{5, 11, 1\} $
- $ K = \{20, 16, 17, 4, 3, 14\} $

#### Problem 2
- $ G \cap H = \{g, h, z\} $
- $ G' \cap H' = \{c, e, a\} $
- $ H' - G' = \{n, x, q\} $
- $ (G - H)' = \{g, h, z, r, b, p, c, e, a\} $

#### Problem 3
- $ A \cup B = \{20, 40, 55, 70, 75, 80, 85, 95\} $
- $ B' = \{15, 20, 25, 50, 55, 95\} $
- $ B - A = \{40, 85\} $
- $ A' \cup B' = \{15, 20, 25, 40, 50, 55, 85, 95\} $

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Let me know if you want these written in boxed format or with explanations for each.