Question

if r is the set of real numbers, q is the set of rational numbers, i is the set of intege through (e) below.
a. ( q cap i )
b. ( s - q )
c. ( r cup s )
d. which of the sets could be a universal set for the other sets?
e. if the universal set is r, how would you describe ( overline{s} )?
a. ( q cap i = square )
b. ( s - q = square )
c. ( r cup s = square )
d. which of the sets could be a universal set for the other sets?
square
e. if the universal set is r, how would you describe ( overline{s} )?
square

Explanation:

Step1: Solve \( Q \cap I \)

Rational numbers \( Q \) include integers, fractions, etc. Integers \( I \) are whole numbers (positive, negative, zero). The intersection of \( Q \) and \( I \) is the set of numbers that are both rational and integers, which is \( I \) (since all integers are rational). So \( Q \cap I = I \).

Step2: Solve \( S - Q \) (Assuming \( S \) is irrational numbers, as it's a common set with \( Q \) in number systems)

The set difference \( S - Q \) is the set of elements in \( S \) but not in \( Q \). If \( S \) is irrational numbers, then \( S - Q \) is the set of irrational numbers (since irrationals are not in \( Q \)). So \( S - Q = S \) (if \( S \) is irrationals).

Step3: Solve \( R \cup S \)

Real numbers \( R \) include rational and irrational numbers. If \( S \) is irrational numbers, then \( R \cup S = R \) (since \( S \) is a subset of \( R \)).

Step4: Determine universal set

A universal set for \( Q \), \( I \), and \( S \) (assuming \( S \) is irrationals) should contain all of them. Real numbers \( R \) contain rationals (\( Q \)), integers (\( I \)), and irrationals (\( S \)), so \( R \) is the universal set.

Step5: Describe \( \overline{S} \) (complement of \( S \) in \( R \))

The complement of \( S \) (irrationals) in \( R \) (reals) is the set of elements in \( R \) not in \( S \), which is the set of rational numbers \( Q \). So \( \overline{S} = Q \).

Answer:

s:
a. \( \boldsymbol{I} \)
b. \( \boldsymbol{S} \) (if \( S \) is irrationals)
c. \( \boldsymbol{R} \)
d. \( \boldsymbol{R} \)
e. \( \boldsymbol{Q} \) (the set of rational numbers)