Inclusion-Exclusion Principle - Intermediate Level: tricky scenarios handling Inclusion-Exclusion Principle INTERMEDIATE

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 12
- Divisors: 2 and 3

Step 1 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊12/2⌋ = 6
These are: 2, 4, 6, ..., 12

Step 2 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊12/3⌋ = 4
These are: 3, 6, 9, ..., 12

Step 3 - Count numbers divisible by BOTH 2 AND 3:
Numbers divisible by LCM(2,3) = 6
Count = ⌊12/6⌋ = 2
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 2) + (divisible by 3) - (divisible by both)
= 6 + 4 - 2
= 8

Visualization (Venn Diagram concept):
- Circle A: divisible by 2 (6 numbers)
- Circle B: divisible by 3 (4 numbers)
- Intersection: divisible by both (2 numbers)
- Union: 8 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 12 - 8 = 4

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 11
- Divisors: 2 and 5

Step 1 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊11/2⌋ = 5
These are: 2, 4, 6, ..., 10

Step 2 - Count numbers divisible by 5:
Numbers divisible by 5 = ⌊11/5⌋ = 2
These are: 5, 10, 15, ..., 10

Step 3 - Count numbers divisible by BOTH 2 AND 5:
Numbers divisible by LCM(2,5) = 10
Count = ⌊11/10⌋ = 1
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 2) + (divisible by 5) - (divisible by both)
= 5 + 2 - 1
= 6

Visualization (Venn Diagram concept):
- Circle A: divisible by 2 (5 numbers)
- Circle B: divisible by 5 (2 numbers)
- Intersection: divisible by both (1 numbers)
- Union: 6 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 11 - 6 = 5

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 8
- Divisors: 2 and 3

Step 1 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊8/2⌋ = 4
These are: 2, 4, 6, ..., 8

Step 2 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊8/3⌋ = 2
These are: 3, 6, 9, ..., 6

Step 3 - Count numbers divisible by BOTH 2 AND 3:
Numbers divisible by LCM(2,3) = 6
Count = ⌊8/6⌋ = 1
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 2) + (divisible by 3) - (divisible by both)
= 4 + 2 - 1
= 5

Visualization (Venn Diagram concept):
- Circle A: divisible by 2 (4 numbers)
- Circle B: divisible by 3 (2 numbers)
- Intersection: divisible by both (1 numbers)
- Union: 5 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 8 - 5 = 3

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 10
- Divisors: 3 and 2

Step 1 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊10/3⌋ = 3
These are: 3, 6, 9, ..., 9

Step 2 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊10/2⌋ = 5
These are: 2, 4, 6, ..., 10

Step 3 - Count numbers divisible by BOTH 3 AND 2:
Numbers divisible by LCM(3,2) = 6
Count = ⌊10/6⌋ = 1
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 3) + (divisible by 2) - (divisible by both)
= 3 + 5 - 1
= 7

Visualization (Venn Diagram concept):
- Circle A: divisible by 3 (3 numbers)
- Circle B: divisible by 2 (5 numbers)
- Intersection: divisible by both (1 numbers)
- Union: 7 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 10 - 7 = 3

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 9
- Divisors: 2 and 3

Step 1 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊9/2⌋ = 4
These are: 2, 4, 6, ..., 8

Step 2 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊9/3⌋ = 3
These are: 3, 6, 9, ..., 9

Step 3 - Count numbers divisible by BOTH 2 AND 3:
Numbers divisible by LCM(2,3) = 6
Count = ⌊9/6⌋ = 1
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 2) + (divisible by 3) - (divisible by both)
= 4 + 3 - 1
= 6

Visualization (Venn Diagram concept):
- Circle A: divisible by 2 (4 numbers)
- Circle B: divisible by 3 (3 numbers)
- Intersection: divisible by both (1 numbers)
- Union: 6 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 9 - 6 = 3

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 11
- Divisors: 2 and 5

Step 1 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊11/2⌋ = 5
These are: 2, 4, 6, ..., 10

Step 2 - Count numbers divisible by 5:
Numbers divisible by 5 = ⌊11/5⌋ = 2
These are: 5, 10, 15, ..., 10

Step 3 - Count numbers divisible by BOTH 2 AND 5:
Numbers divisible by LCM(2,5) = 10
Count = ⌊11/10⌋ = 1
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 2) + (divisible by 5) - (divisible by both)
= 5 + 2 - 1
= 6

Visualization (Venn Diagram concept):
- Circle A: divisible by 2 (5 numbers)
- Circle B: divisible by 5 (2 numbers)
- Intersection: divisible by both (1 numbers)
- Union: 6 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 11 - 6 = 5

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 8
- Divisors: 2 and 3

Step 1 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊8/2⌋ = 4
These are: 2, 4, 6, ..., 8

Step 2 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊8/3⌋ = 2
These are: 3, 6, 9, ..., 6

Step 3 - Count numbers divisible by BOTH 2 AND 3:
Numbers divisible by LCM(2,3) = 6
Count = ⌊8/6⌋ = 1
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 2) + (divisible by 3) - (divisible by both)
= 4 + 2 - 1
= 5

Visualization (Venn Diagram concept):
- Circle A: divisible by 2 (4 numbers)
- Circle B: divisible by 3 (2 numbers)
- Intersection: divisible by both (1 numbers)
- Union: 5 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 8 - 5 = 3

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 10
- Divisors: 2 and 5

Step 1 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊10/2⌋ = 5
These are: 2, 4, 6, ..., 10

Step 2 - Count numbers divisible by 5:
Numbers divisible by 5 = ⌊10/5⌋ = 2
These are: 5, 10, 15, ..., 10

Step 3 - Count numbers divisible by BOTH 2 AND 5:
Numbers divisible by LCM(2,5) = 10
Count = ⌊10/10⌋ = 1
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 2) + (divisible by 5) - (divisible by both)
= 5 + 2 - 1
= 6

Visualization (Venn Diagram concept):
- Circle A: divisible by 2 (5 numbers)
- Circle B: divisible by 5 (2 numbers)
- Intersection: divisible by both (1 numbers)
- Union: 6 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 10 - 6 = 4

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 11
- Divisors: 3 and 5

Step 1 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊11/3⌋ = 3
These are: 3, 6, 9, ..., 9

Step 2 - Count numbers divisible by 5:
Numbers divisible by 5 = ⌊11/5⌋ = 2
These are: 5, 10, 15, ..., 10

Step 3 - Count numbers divisible by BOTH 3 AND 5:
Numbers divisible by LCM(3,5) = 15
Count = ⌊11/15⌋ = 0
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 3) + (divisible by 5) - (divisible by both)
= 3 + 2 - 0
= 5

Visualization (Venn Diagram concept):
- Circle A: divisible by 3 (3 numbers)
- Circle B: divisible by 5 (2 numbers)
- Intersection: divisible by both (0 numbers)
- Union: 5 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 11 - 5 = 6

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 12
- Divisors: 2 and 5

Step 1 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊12/2⌋ = 6
These are: 2, 4, 6, ..., 12

Step 2 - Count numbers divisible by 5:
Numbers divisible by 5 = ⌊12/5⌋ = 2
These are: 5, 10, 15, ..., 10

Step 3 - Count numbers divisible by BOTH 2 AND 5:
Numbers divisible by LCM(2,5) = 10
Count = ⌊12/10⌋ = 1
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 2) + (divisible by 5) - (divisible by both)
= 6 + 2 - 1
= 7

Visualization (Venn Diagram concept):
- Circle A: divisible by 2 (6 numbers)
- Circle B: divisible by 5 (2 numbers)
- Intersection: divisible by both (1 numbers)
- Union: 7 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 12 - 7 = 5

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 8
- Divisors: 3 and 2

Step 1 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊8/3⌋ = 2
These are: 3, 6, 9, ..., 6

Step 2 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊8/2⌋ = 4
These are: 2, 4, 6, ..., 8

Step 3 - Count numbers divisible by BOTH 3 AND 2:
Numbers divisible by LCM(3,2) = 6
Count = ⌊8/6⌋ = 1
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 3) + (divisible by 2) - (divisible by both)
= 2 + 4 - 1
= 5

Visualization (Venn Diagram concept):
- Circle A: divisible by 3 (2 numbers)
- Circle B: divisible by 2 (4 numbers)
- Intersection: divisible by both (1 numbers)
- Union: 5 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 8 - 5 = 3

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 12
- Divisors: 2 and 3

Step 1 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊12/2⌋ = 6
These are: 2, 4, 6, ..., 12

Step 2 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊12/3⌋ = 4
These are: 3, 6, 9, ..., 12

Step 3 - Count numbers divisible by BOTH 2 AND 3:
Numbers divisible by LCM(2,3) = 6
Count = ⌊12/6⌋ = 2
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 2) + (divisible by 3) - (divisible by both)
= 6 + 4 - 2
= 8

Visualization (Venn Diagram concept):
- Circle A: divisible by 2 (6 numbers)
- Circle B: divisible by 3 (4 numbers)
- Intersection: divisible by both (2 numbers)
- Union: 8 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 12 - 8 = 4

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 10
- Divisors: 3 and 5

Step 1 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊10/3⌋ = 3
These are: 3, 6, 9, ..., 9

Step 2 - Count numbers divisible by 5:
Numbers divisible by 5 = ⌊10/5⌋ = 2
These are: 5, 10, 15, ..., 10

Step 3 - Count numbers divisible by BOTH 3 AND 5:
Numbers divisible by LCM(3,5) = 15
Count = ⌊10/15⌋ = 0
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 3) + (divisible by 5) - (divisible by both)
= 3 + 2 - 0
= 5

Visualization (Venn Diagram concept):
- Circle A: divisible by 3 (3 numbers)
- Circle B: divisible by 5 (2 numbers)
- Intersection: divisible by both (0 numbers)
- Union: 5 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 10 - 5 = 5

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 8
- Divisors: 3 and 5

Step 1 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊8/3⌋ = 2
These are: 3, 6, 9, ..., 6

Step 2 - Count numbers divisible by 5:
Numbers divisible by 5 = ⌊8/5⌋ = 1
These are: 5, 10, 15, ..., 5

Step 3 - Count numbers divisible by BOTH 3 AND 5:
Numbers divisible by LCM(3,5) = 15
Count = ⌊8/15⌋ = 0
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 3) + (divisible by 5) - (divisible by both)
= 2 + 1 - 0
= 3

Visualization (Venn Diagram concept):
- Circle A: divisible by 3 (2 numbers)
- Circle B: divisible by 5 (1 numbers)
- Intersection: divisible by both (0 numbers)
- Union: 3 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 8 - 3 = 5

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 8
- Divisors: 3 and 2

Step 1 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊8/3⌋ = 2
These are: 3, 6, 9, ..., 6

Step 2 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊8/2⌋ = 4
These are: 2, 4, 6, ..., 8

Step 3 - Count numbers divisible by BOTH 3 AND 2:
Numbers divisible by LCM(3,2) = 6
Count = ⌊8/6⌋ = 1
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 3) + (divisible by 2) - (divisible by both)
= 2 + 4 - 1
= 5

Visualization (Venn Diagram concept):
- Circle A: divisible by 3 (2 numbers)
- Circle B: divisible by 2 (4 numbers)
- Intersection: divisible by both (1 numbers)
- Union: 5 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 8 - 5 = 3

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 9
- Divisors: 2 and 5

Step 1 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊9/2⌋ = 4
These are: 2, 4, 6, ..., 8

Step 2 - Count numbers divisible by 5:
Numbers divisible by 5 = ⌊9/5⌋ = 1
These are: 5, 10, 15, ..., 5

Step 3 - Count numbers divisible by BOTH 2 AND 5:
Numbers divisible by LCM(2,5) = 10
Count = ⌊9/10⌋ = 0
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 2) + (divisible by 5) - (divisible by both)
= 4 + 1 - 0
= 5

Visualization (Venn Diagram concept):
- Circle A: divisible by 2 (4 numbers)
- Circle B: divisible by 5 (1 numbers)
- Intersection: divisible by both (0 numbers)
- Union: 5 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 9 - 5 = 4

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 12
- Divisors: 2 and 3

Step 1 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊12/2⌋ = 6
These are: 2, 4, 6, ..., 12

Step 2 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊12/3⌋ = 4
These are: 3, 6, 9, ..., 12

Step 3 - Count numbers divisible by BOTH 2 AND 3:
Numbers divisible by LCM(2,3) = 6
Count = ⌊12/6⌋ = 2
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 2) + (divisible by 3) - (divisible by both)
= 6 + 4 - 2
= 8

Visualization (Venn Diagram concept):
- Circle A: divisible by 2 (6 numbers)
- Circle B: divisible by 3 (4 numbers)
- Intersection: divisible by both (2 numbers)
- Union: 8 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 12 - 8 = 4

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 8
- Divisors: 2 and 3

Step 1 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊8/2⌋ = 4
These are: 2, 4, 6, ..., 8

Step 2 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊8/3⌋ = 2
These are: 3, 6, 9, ..., 6

Step 3 - Count numbers divisible by BOTH 2 AND 3:
Numbers divisible by LCM(2,3) = 6
Count = ⌊8/6⌋ = 1
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 2) + (divisible by 3) - (divisible by both)
= 4 + 2 - 1
= 5

Visualization (Venn Diagram concept):
- Circle A: divisible by 2 (4 numbers)
- Circle B: divisible by 3 (2 numbers)
- Intersection: divisible by both (1 numbers)
- Union: 5 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 8 - 5 = 3

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 8
- Divisors: 3 and 2

Step 1 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊8/3⌋ = 2
These are: 3, 6, 9, ..., 6

Step 2 - Count numbers divisible by 2:
Numbers divisible by 2 = ⌊8/2⌋ = 4
These are: 2, 4, 6, ..., 8

Step 3 - Count numbers divisible by BOTH 3 AND 2:
Numbers divisible by LCM(3,2) = 6
Count = ⌊8/6⌋ = 1
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 3) + (divisible by 2) - (divisible by both)
= 2 + 4 - 1
= 5

Visualization (Venn Diagram concept):
- Circle A: divisible by 3 (2 numbers)
- Circle B: divisible by 2 (4 numbers)
- Intersection: divisible by both (1 numbers)
- Union: 5 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 8 - 5 = 3

Step-by-Step Solution:

Concept: Inclusion-Exclusion Principle - when counting elements in the union of sets, add individual counts and subtract overcounted intersections.

Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

Given:
- Range: 1 to 8
- Divisors: 3 and 5

Step 1 - Count numbers divisible by 3:
Numbers divisible by 3 = ⌊8/3⌋ = 2
These are: 3, 6, 9, ..., 6

Step 2 - Count numbers divisible by 5:
Numbers divisible by 5 = ⌊8/5⌋ = 1
These are: 5, 10, 15, ..., 5

Step 3 - Count numbers divisible by BOTH 3 AND 5:
Numbers divisible by LCM(3,5) = 15
Count = ⌊8/15⌋ = 0
(These are counted twice in steps 1 and 2)

Step 4 - Apply Inclusion-Exclusion:
Total = (divisible by 3) + (divisible by 5) - (divisible by both)
= 2 + 1 - 0
= 3

Visualization (Venn Diagram concept):
- Circle A: divisible by 3 (2 numbers)
- Circle B: divisible by 5 (1 numbers)
- Intersection: divisible by both (0 numbers)
- Union: 3 numbers

Key Principle: Subtract the intersection to avoid double counting.

Extension to Three Sets:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Common Application: Finding numbers NOT divisible by either = 8 - 3 = 5