Question

the universal set is the set of lower case letters from a to n. the sets r, s, and t are defined as indicated in the venn diagram below: venn diagram with circles s, t, r; letters a, i in s; b in t; c, e, n in s∩r; g in r∩t; j, k, f in r; h, l, d, m in universal set outside r, s, t. write out the following sets in roster notation. ( r cap t = { } ) ( t cup s = { } ) ( r = { } ) ( s cap t = { } ) ( s = { } ) ( r cap s cap t = { } )

Explanation:

1. Solve \( R \cap T \)

Step1: Identify elements in R and T

Set \( R \) elements: \( \{c, e, n, j, k, f, g\} \) (from the Venn diagram, the circle labeled R). Set \( T \) elements: \( \{c, b, g, n, e\} \) (from the circle labeled T).

Step2: Find intersection (common elements)

Common elements in both \( R \) and \( T \) are \( c, e, n, g \).

Step1: Identify elements in T and S

Set \( T \) elements: \( \{c, b, g, n, e\} \). Set \( S \) elements: \( \{a, i, c, e, n\} \) (from the circle labeled S).

Step2: Find union (all elements from both)

Combine elements, removing duplicates: \( a, i, c, e, n, b, g \).

Step1: Identify universal set

Universal set \( U = \{a, b, c, d, e, f, g, h, i, j, k, l, m, n\} \) (lowercase letters from a to n).

Step2: Identify elements not in R

Elements in \( U \) but not in \( R \) ( \( R = \{c, e, n, j, k, f, g\} \) ) are \( a, b, d, h, i, l, m \).

Answer:

\( R \cap T = \{c, e, n, g\} \)

2. Solve \( T \cup S \)