Question

construct a venn diagram illustrating the sets below. u={a, b, c, d, e, f, g} x={b, c, d, e, f, g} z={b, c, d, f}

Explanation:

Step1: Analyze the universal set and subsets

The universal set \( U = \{a, b, c, d, e, f, g\} \), set \( X=\{b, c, d, e, f, g\} \), set \( Z = \{b, c, d, f\} \). First, find the elements in only \( X \), only \( Z \), the intersection of \( X \) and \( Z \), and the element in \( U \) but not in \( X \) or \( Z \).

• Element in \( U \) but not in \( X \) or \( Z \): \( a \) (since \( X \) contains \( b - g \), so \( a \) is outside both circles).
• Intersection of \( X \) and \( Z \) (\( X\cap Z \)): Elements common to both \( X \) and \( Z \), which are \( \{b, c, d, f\} \) (since \( Z \) is a subset of \( X \) for these elements? Wait, \( Z \) is \( \{b,c,d,f\} \) and \( X \) is \( \{b,c,d,e,f,g\} \), so \( X\cap Z=\{b,c,d,f\} \)).
• Elements in \( X \) but not in \( Z \): \( X - Z=\{e, g\} \) (since \( X \) has \( e, g \) which are not in \( Z \)).
• Elements in \( Z \) but not in \( X \): Since \( Z \) is a subset of \( X \) (all elements of \( Z \) are in \( X \)), there are no elements here.

Step2: Place elements in the Venn diagram

• Outside both circles (in \( U \) only): \( a \).
• In the intersection of \( X \) and \( Z \): \( b, c, d, f \).
• In \( X \) only (not in \( Z \)): \( e, g \).
• In \( Z \) only: None (since \( Z \subseteq X \)).

So the Venn diagram should have:

• The region outside both circles (labeled \( U \) but not \( X \) or \( Z \)) with \( a \).
• The overlapping region ( \( X\cap Z \)) with \( b, c, d, f \).
• The region of \( X \) not overlapping with \( Z \) with \( e, g \).
• The region of \( Z \) not overlapping with \( X \) is empty.

Answer:

To construct the Venn diagram:

• Place \( a \) outside both the red (X) and blue (Z) circles (in the universal set \( U \) region not in \( X \) or \( Z \)).
• Place \( b, c, d, f \) in the overlapping region of the red (X) and blue (Z) circles ( \( X\cap Z \) ).
• Place \( e, g \) in the region of the red (X) circle that does not overlap with the blue (Z) circle ( \( X - Z \) ).
• The region of the blue (Z) circle that does not overlap with the red (X) circle is empty (no elements).