Sets and Venn Diagrams (A) Worksheet | Printable PDF Worksheets - Free Printable

Here is the complete solution to the "Sets and Venn Diagrams (A)" worksheet.

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Section A: Correctly list all the data in the sets.

1. Even numbers between 1 and 10 = {2, 4, 6, 8, 10}
* *Explanation:* Even numbers are integers divisible by 2. Between 1 and 10 (not including 1 and 10 themselves? The phrasing "between 1 and 10" can be ambiguous, but in standard math contexts for such problems, it usually means from 2 to 9. However, looking at the provided sets in Section B (e.g., `Even = {2, 4, 6, 8, 10}`), it's clear the problem intends to include 10. So, the even numbers are 2, 4, 6, 8, and 10.*

2. Odd numbers between 5 and 15 = {7, 9, 11, 13}
* *Explanation:* Odd numbers are integers not divisible by 2. "Between 5 and 15" typically excludes 5 and 15 themselves. So, we list the odd numbers starting from the next integer after 5, which is 6 (even), then 7 (odd), 8 (even), 9 (odd), 10 (even), 11 (odd), 12 (even), 13 (odd), 14 (even). The next number is 15, which is excluded. Therefore, the set is {7, 9, 11, 13}.*

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Section B: Sort the data and choose the correct statement below, A, B, or C.

#### First Venn Diagram: Even and Odd
* Given Sets:
* Even = {2, 4, 6, 8, 10}
* Odd = {1, 3, 5, 7, 9}
* Question: Choose the correct statement about `Even ∪ Odd`.
* `∪` means Union (all elements in either set).
* `Even ∪ Odd` = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
* Correct Answer: A
* *Explanation:* Statement A correctly lists the union of the two sets. Statement B is wrong because the union is not empty. Statement C only lists a subset of the union.

#### Second Venn Diagram: Even and Prime
* Given Sets:
* Even = {2, 4, 6, 8, 10}
* Prime = {2, 3, 5, 7} (Prime numbers are greater than 1 with no positive divisors other than 1 and themselves).
* Question: Choose the correct statement about `Even ∩ Prime`.
* `∩` means Intersection (elements common to both sets).
* The only number that is both even and prime is 2.
* `Even ∩ Prime` = {2}
* Correct Answer: B
* *Explanation:* Statement B correctly identifies the intersection as {2}. Statement A includes 4 and 7, which are not in both sets. Statement C includes 6 and 8, which are not prime.

#### Third Venn Diagram: Even and Square
* Given Sets:
* Even = {2, 4, 6, 8, 10, 12, 14, 16}
* Square = {4, 9, 16, 25} (These are perfect squares: 2²=4, 3²=9, 4²=16, 5²=25).
* Question: Choose the correct statement about `Even ∩ Square`.
* `∩` means Intersection (elements common to both sets).
* We need numbers that are both even AND perfect squares.
* From the Square set {4, 9, 16, 25}, check which are even:
* 4 is even.
* 9 is odd.
* 16 is even.
* 25 is odd.
* So, `Even ∩ Square` = {4, 16}
* Correct Answer: C
* *Explanation:* Statement C correctly identifies the intersection as {4, 16}. Statement A includes 2 and 6, which are even but not perfect squares. Statement B describes the Union (`Even ∪ Square`), not the Intersection.

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Section C

#### 1) Chess Board (Black pieces B and Rooks R)

* Venn Diagram Numbers:
* Only Black (not Rooks): 14
* Black AND Rooks (intersection): 2
* Only Rooks (not Black): 2
* Total pieces shown: 14 + 2 + 2 = 18

* Questions:
1. How many pieces are black and rooks?
* *Answer:* 2
* *Explanation:* This is the number in the intersection of the two circles, representing pieces that belong to both sets.

2. Describe B ∪ R in words.
* *Answer:* The set of all pieces that are either black or rooks or both.
* *Explanation:* `B ∪ R` (Union) includes everything in the Black circle, everything in the Rooks circle, and everything in their overlap. It represents any piece that has at least one of the properties: being black OR being a rook.

3. How many pieces are B'?
* *Answer:* 4
* *Explanation:* `B'` (B complement) means "not Black". These are the pieces that are outside the Black circle. In the diagram, this is the region containing only Rooks (2 pieces) plus any pieces outside both circles. The diagram shows a "14" outside the entire Venn diagram, which likely represents pieces that are neither black nor rooks. However, the question asks for `B'`, which is everything *not* in set B. This includes the "Only Rooks" part (2) and the "Neither" part (14). So, `B'` = 2 + 14 = 16.
* *Correction:* Looking at the diagram again, the "14" written *outside* the entire Venn diagram is likely meant to represent the total number of pieces that are *neither* Black *nor* Rooks. The number "14" *inside* the Black circle (non-overlapping part) is the count of pieces that are Black but not Rooks. The "2" in the overlap is Black AND Rooks. The "2" in the Rooks-only part is Rooks but not Black.
* Therefore, `B'` (pieces that are NOT Black) = Pieces that are Only Rooks + Pieces that are Neither = 2 + 14 = 16.
* *Note:* The initial answer of 4 was incorrect. The correct answer is 16.

#### 2) Playing Cards (Clubs C and Royal cards R)

* Venn Diagram Numbers:
* Only Clubs (not Royal): 10
* Clubs AND Royal (intersection): 3
* Only Royal (not Clubs): 9
* Total cards shown: 10 + 3 + 9 = 22
* Total cards in pack (given outside diagram): 30

* Questions:
* C ∪ R (Union: Clubs or Royal or both)
* *Calculation:* Only Clubs + Only Royal + Both = 10 + 9 + 3 = 22
* *Answer:* 22
* C' ∪ R (Complement of Clubs OR Royal)
* `C'` means "Not Clubs". `C' ∪ R` means "Not Clubs OR Royal".
* This includes:
* Pieces that are Not Clubs (which includes Only Royal and Neither).
* Plus pieces that are Royal (but we already included them in "Not Clubs" if they weren't Clubs, so we don't double count).
* Alternatively, think of what is *not* included: Only things that are Clubs AND Not Royal are excluded.
* The only region excluded from `C' ∪ R` is the "Only Clubs" part (10).
* So, `C' ∪ R` = Total cards - Only Clubs = 30 - 10 = 20.
* *Answer:* 20
* C ∪ R' (Clubs OR Complement of Royal)
* `R'` means "Not Royal". `C ∪ R'` means "Clubs OR Not Royal".
* This includes:
* All Clubs (Only Clubs + Both) = 10 + 3 = 13.
* Plus all Not Royal cards that are not already counted (i.e., Not Royal and Not Clubs).
* The region "Not Royal and Not Clubs" is the part outside both circles. Total cards = 30. Cards inside C or R = 22. So, cards outside both = 30 - 22 = 8.
* So, `C ∪ R'` = (Only Clubs + Both) + (Neither) = 10 + 3 + 8 = 21.
* *Answer:* 21

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Final Answers Summary

Section A:
* Even numbers between 1 and 10 = {2, 4, 6, 8, 10}
* Odd numbers between 5 and 15 = {7, 9, 11, 13}

Section B:
* First Diagram (Even ∪ Odd): A
* Second Diagram (Even ∩ Prime): B
* Third Diagram (Even ∩ Square): C

Section C:
* 1) Chess Board:
1. Black and Rooks: 2
2. B ∪ R: The set of all pieces that are either black or rooks or both.
3. B': 16 *(Corrected from initial error)*
* 2) Playing Cards:
* C ∪ R: 22
* C' ∪ R: 20
* C ∪ R': 21