Question

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

Explanation:

Step1: Find \( Y \cap Z \)

The intersection of \( Y \) and \( Z \) is the set of elements common to both \( Y \) and \( Z \). Given \( Y = \{c, d, e\} \) and \( Z = \{b, c, e, f, g\} \), the common elements are \( c \) and \( e \). So \( Y \cap Z=\{c, e\} \).

Step2: Find elements only in \( Y \)

Elements in \( Y \) but not in \( Z \): From \( Y = \{c, d, e\} \), remove the elements of \( Y \cap Z \) (\( c, e \)). So we get \( d \).

Step3: Find elements only in \( Z \)

Elements in \( Z \) but not in \( Y \): From \( Z = \{b, c, e, f, g\} \), remove the elements of \( Y \cap Z \) (\( c, e \)). So we get \( b, f, g \).

Step4: Find elements in \( U \) but not in \( Y \) or \( Z \)

The universal set \( U = \{a, b, c, d, e, f, g\} \). Remove all elements in \( Y \) or \( Z \) (i.e., \( b, c, d, e, f, g \)) from \( U \). We are left with \( a \).

Step5: Place elements in Venn diagram

• Region only \( Y \): \( d \)
• Region \( Y \cap Z \): \( c, e \)
• Region only \( Z \): \( b, f, g \)
• Region outside both \( Y \) and \( Z \) (in \( U \)): \( a \)

Answer:

To construct the Venn diagram:

• Inside circle \( Y \) (but not overlapping with \( Z \)): place \( d \)
• In the overlapping region of \( Y \) and \( Z \): place \( c, e \)
• Inside circle \( Z \) (but not overlapping with \( Y \)): place \( b, f, g \)
• Outside both circles (within the universal set \( U \)): place \( a \)

The Venn diagram structure (the first option with two overlapping circles) is used, and elements are placed as per the above breakdown.