Question

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

Explanation:

Step1: Find \( Y \cap Z \)

First, identify the intersection of sets \( Y \) and \( Z \). The elements common to both \( Y = \{d, f, h\} \) and \( Z = \{c, e, f, g, h, i\} \) are \( f \) and \( h \). So \( Y \cap Z=\{f, h\} \).

Step2: Find elements only in \( Y \)

Elements in \( Y \) but not in \( Z \): From \( Y = \{d, f, h\} \), remove the intersection elements (\( f, h \)). So only in \( Y \): \( \{d\} \).

Step3: Find elements only in \( Z \)

Elements in \( Z \) but not in \( Y \): From \( Z = \{c, e, f, g, h, i\} \), remove the intersection elements (\( f, h \)). So only in \( Z \): \( \{c, e, g, i\} \).

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

Universal set \( U = \{a, b, c, d, e, f, g, h, i\} \). Remove elements in \( Y \) or \( Z \) (i.e., \( d, f, h, c, e, g, i \)). Remaining elements: \( \{a, b\} \).

Step5: Place elements in Venn diagram

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

Answer:

To construct the Venn diagram:

• In the red circle (set \( Y \)) only: place \( d \).
• In the overlapping region of red ( \( Y \)) and blue ( \( Z \)) circles: place \( f, h \).
• In the blue circle (set \( Z \)) only: place \( c, e, g, i \).
• In the area inside the universal set \( U \) but outside both circles: place \( a, b \).