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It is given that \( \mathrm{n}(C)=h \) and \( \mathrm{n}(D)=k \). (a) If \( D \subset C \), find an expression, in terms of \( h \) and/or \( k \), for (i) \( \mathrm{n}(C \cap D) \), (ii) \( \mathrm{n}(C \cup D) \). (b) If \( D \subseteq C \), find an expression, in terms of \( h \) and/or \( k \), for the minimum value of \( \mathrm{n}\left(C \cap D^{\prime}\right) \).

Ask by Ward Christensen. in Singapore
Dec 16,2024

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(i) \( \mathrm{n}(C \cap D) = k \) (ii) \( \mathrm{n}(C \cup D) = h \) (b) The minimum value of \( \mathrm{n}(C \cap D') \) is \( 0 \).

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Let's delve into the problem step by step. (a) Given that \( D \subset C \) and denoting \( \mathrm{n}(C) = h \) and \( \mathrm{n}(D) = k \): (i) Since \( D \) is a subset of \( C \), all elements of \( D \) are also in \( C \). Thus, the number of elements in the intersection of \( C \) and \( D \) can be expressed as: \[ \mathrm{n}(C \cap D) = \mathrm{n}(D) = k. \] (ii) The union of \( C \) and \( D \) when \( D \subset C \) simply equals \( C \), hence: \[ \mathrm{n}(C \cup D) = \mathrm{n}(C) = h. \] (b) When \( D \subseteq C \), we want to find the minimum value of \( \mathrm{n}(C \cap D^{\prime}) \), where \( D^{\prime} \) is the complement of \( D \) within \( C \). Since \( D \) can contain at most \( k \) elements, the remaining elements of \( C \) (which are not in \( D \)) contribute to \( C \cap D^{\prime} \). Therefore, the size of \( C \cap D^{\prime} \) can be expressed as: \[ \mathrm{n}(C \cap D^{\prime}) = \mathrm{n}(C) - \mathrm{n}(D) = h - k. \] This value represents the immediate complement of \( D \) in \( C \). Thus, the minimum value of \( \mathrm{n}(C \cap D^{\prime}) \) is \( h - k \).

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