Question

question 6 of 30
for each pair of statements, choose the one that is true.
question 7 of 30
in the braces below, list all subsets of the set {3, 4}.
write each subset in your list in roster form. if there is more than one subset in your list, separate them with commas. if you need the empty set in your list, use the symbol ∅.
question 8 of 30
let the set a be defined as follows.
a={m,g,r,y,d,c}
(a) find the total number of subsets of a.
(b) find the total number of proper subsets of a.
question 9 of 30
diane, a college freshman, is deciding on classes for her first - term. she can choose to take some or all of the seven classes from the list below. she must take at least one of them.
astronomy, biology, chemistry, dance, economics, geography, history
in how many different ways can diane choose the classes to take? (the order of the classes chosen doesnt matter.)

Explanation:

Question 7

Step1: Recall set - element and subset relations

The symbol $\in$ means an element belongs to a set, and $\subseteq$ means a set is a subset of another set.

• (a) $\{4\}\subseteq\{5,6,7\}$ is false because $4$ is not an element of $\{5,6,7\}$.
• (b) $\{4\}\in\{4,5,6\}$ is false. $\{4\}$ is a set, and the elements of $\{4,5,6\}$ are numbers, not sets.
• (c) $\{k\}\subseteq\{h,k\}$ is true since every element of $\{k\}$ (which is just $k$) is also an element of $\{h,k\}$.
• (d) $k\subseteq\{h,k\}$ is false. $k$ is an element, not a set, so the subset relation does not apply.

Question 8

Step1: Recall the formula for the number of subsets

If a set $A$ has $n$ elements, the number of subsets of $A$ is $2^n$, and the number of proper - subsets of $A$ is $2^n - 1$.
The set $A=\{m,g,r,y,d,c\}$ has $n = 6$ elements.

Step2: Calculate the number of subsets

The number of subsets of $A$ is $2^n=2^6 = 64$.

Step3: Calculate the number of proper - subsets

The number of proper - subsets of $A$ is $2^n-1=2^6 - 1=63$.

Question 9

Step1: Use the principle of inclusion - exclusion (or consider cases)

The set of classes has $n = 7$ elements. The total number of non - empty subsets (ways to choose at least one class) is the total number of subsets minus the empty set.
The total number of subsets of a set with $n$ elements is $2^n$. Here $n = 7$, so the total number of subsets is $2^7=128$.
The number of ways to choose at least one class is $2^7-1 = 127$.

Answer:

Question 7: c
Question 8:
(a) 64
(b) 63
Question 9: 127