Q1Discrete Mathematics Structure

Question

Out of 250 failed students, 128 failed in Maths, 87 in Physics and 134 in aggregate, 31 failed in Maths and Physics, 54 failed in aggregate and in Maths, 30 failed in aggregate and in Physics. Find how many candidates failed : (i) In all three subjects (ii) In Maths not in Physics (iii) In aggregate but not in Maths (iv) In Physics but not in aggregate or Maths (v) In the aggregate or in Maths, but not in Physics

Answer 1

Let $M, P, A$ be sets of students failed in Maths, Physics, and Aggregate. $∣ U ∣ = 250$ .

Given: $∣ M ∣ = 128, ∣ P ∣ = 87, ∣ A ∣ = 134, ∣ M \cap P ∣ = 31, ∣ M \cap A ∣ = 54, ∣ P \cap A ∣ = 30$ . Let $X = ∣ M \cap P \cap A ∣$ .

Using $∣ M \cup P \cup A ∣ = ∣ M ∣ + ∣ P ∣ + ∣ A ∣ - (∣ M \cap P ∣ + ∣ M \cap A ∣ + ∣ P \cap A ∣) + X$

$250 = 128 + 87 + 134 - (31 + 54 + 30) + X = 349 - 115 + X = 234 + X ⟹ X = 16$ .

(i) All three: 16.
(ii) Maths not in Physics: $128 - 31 = 97$ .
(iii) Aggregate but not in Maths: $134 - 54 = 80$ .
(iv) Physics but not in aggregate or Maths: $∣ P ∣ - ∣ P \cap (A \cup M) ∣ = 87 - (31 + 30 - 16) = 87 - 45 = 42$ (Recalculated).
(v) Aggregate or Maths, but not in Physics: $∣ A \cup M ∣ - ∣ (A \cup M) \cap P ∣ = (128 + 134 - 54) - (31 + 30 - 16) = 208 - 45 = 163$ .

Q1Discrete Mathematics Structure

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