Sets, Venn Diagrams and Variation

Why this topic matters

WAEC and NECO set a Venn diagram word problem and a variation question almost every year, in both Paper 1 and Paper 2. Both follow fixed methods, so they are reliable marks once you know the steps.

Set language

• ∈ means "is an element of"; ⊂ means "is a subset of".
• A ∪ B (union) is everything in A or B or both.
• A ∩ B (intersection) is what is in A and B.
• A′ (complement) is everything not in A.
• ∅ or { } is the empty set; U or ξ is the universal set.
• n(A) is the number of elements in A.

Subsets: a set with k elements has 2ᵏ subsets (including ∅ and the set itself), and 2ᵏ − 1 proper subsets. For example, B = {3, 6, 9} has 2³ = 8 subsets.

Reading Venn diagram regions

Number the four regions: ① A only, ② both, ③ B only, ④ neither. Then:

• A ∪ B = ①②③
• A ∩ B = ②
• A′ = ③④
• (A ∪ B)′ = ④
• A ∩ B′ = ① (A only)

Two-set word problems

Key formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Always check that all the regions add up to n(U).

Three-set word problems: work from the centre outwards

1. Centre: all three = 3.
2. Pairs only: C and P = 7 − 3 = 4; C and B = 6 − 3 = 3; P and B = 5 − 3 = 2.
3. One subject only: C = 20 − 4 − 3 − 3 = 10; P = 18 − 4 − 2 − 3 = 9; B = 16 − 3 − 2 − 3 = 8.
4. Total inside the circles = 10 + 9 + 8 + 4 + 3 + 2 + 3 = 39, so none = 40 − 39 = 1.

Variation

Each type of variation has its own equation:

• Direct: y ∝ x, so y = kx
• Direct (square): y ∝ x², so y = kx²
• Inverse: y ∝ 1/x, so y = k ÷ x
• Inverse (root): y ∝ 1/√x, so y = k ÷ √x
• Joint: z ∝ xy, so z = kxy
• Combined: y ∝ x and inversely as z², so y = kx ÷ z²
• Partial: partly fixed and partly varying, so C = a + bn

The 4-step method:

1. Write the relation with ∝.
2. Replace ∝ with "= k".
3. Use the given pair of values to find k.
4. Write out the law, then substitute.

What the graphs look like:

• Direct: a straight line through the origin.
• Square: a parabola.
• Inverse: a curve that never touches the axes.
• Partial: a straight line that does not pass through the origin.

Worked examples

1. Direct (square): y ∝ x², and y = 18 when x = 3. Then 18 = 9k, so k = 2 and the law is y = 2x². When x = 5, y = 50.
2. Inverse: P varies inversely as Q, and P = 6 when Q = 4. Then k = 24 and the law is P = 24 ÷ Q. When P = 8, Q = 3.
3. Joint: z varies jointly as x and y, and z = 12 when x = 2 and y = 3. Then 12 = 6k, so k = 2. When x = 5 and y = 4, z = 40.
4. Partial (a WAEC favourite): the cost C of hiring a hall is partly fixed and partly varies as the number of guests n. C = ₦25,000 for 50 guests and ₦40,000 for 100 guests. Then a + 50b = 25,000 and a + 100b = 40,000, so 50b = 15,000, giving b = 300 and a = 10,000. For 80 guests, C = 10,000 + 300 × 80 = ₦34,000.

Top exam traps

• Ignoring "only" in Venn problems. Fill the middle first.
• Forgetting the "neither" region outside the circles.
• Not writing down the law (e.g. y = 2x²) before substituting.
• Confusing direct and inverse: in inverse variation, x × y stays constant.