This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems. Problem 19: Let E be the set of students who take Economics and H be the set of students who take Home Management. Given: Total students n(U) = 45 Students taking Economics n(E) = 25 Students taking Home Management n(H) = 21 Students taking neither subject n(E H)' = 3 Step 1: Find the number of students who take at least one subject. n(E H) = n(U) - n(E H)' n(E H) = 45 - 3 = 42 Step 2: Use the Principle of Inclusion-Exclusion to find the number of students who take both subjects. n(E H) = n(E) + n(H) - n(E H) 42 = 25 + 21 - n(E H) 42 = 46 - n(E H) n(E H) = 46 - 42 = 4 Step 3: Find the number of students who take Economics only. n(Economics only) = n(E) - n(E H) n(Economics only) = 25 - 4 = 21 The correct option is C. The number of students who take Economics only is C. 21. Problem 20: Given set P = \2, 4, 6\. A set A is a subset of P if every element in A is also an element in P. A. \2, 2, 4\ is equivalent to \2, 4\. Both 2 and 4 are in P. So, \2, 2, 4\ is a subset of P. B. \2, 3, 4\. The element 3 is in this set, but 3 is not an element of P. Therefore, \2, 3, 4\ is not a subset of P. C. \2, 4, 6\. All elements (2, 4, 6) are in P. So, \2, 4, 6\ is a subset of P. D. . The empty set is a subset of every set. So, is a subset of P. The correct option is B. The set that is not a subset of P is B. \2, 3, 4\. Problem 21: We need to find the equivalent expression for (M' N)'. Using De Morgan's Laws, which state that (A B)' = A' B' and (A B)' = A' B'. Also, (A')' = A. Apply De Morgan's Law to (M' N)': (M' N)' = (M')' N' (M' N)' = M N' The correct option is C. (M' N)' is equivalent to C. (M N'). Problem 22: Let S be the set of teachers who teach pure Science and T be the set of teachers who teach Social Science. Given: Total teachers n(U) = 65 Teachers teaching pure Science n(S) = 45 Teachers teaching Social Science n(T) = 32 Teachers who do not teach any subject n(S T)' = 7 Step 1: Find the number of teachers who teach at least one subject. n(S T) = n(U) - n(S T)' n(S T) = 65 - 7 = 58 Step 2: Use the Principle of Inclusion-Exclusion to find the number of teachers who teach both subjects. n(S T) = n(S) + n(T) - n(S T) 58 = 45 + 32 - n(S T) 58 = 77 - n(S T) n(S T) = 77 - 58 = 19 The correct option is D. The number of teachers who teach both subjects is D. 19.