Question

lesson 2

1. list the elements of each of the following sets.

a) p = {prime numbers less than 25}
2, 3, 5, 7, 11, 13, 17, 19, 23
b) m = {positive multiples of 3 that are less than 25}
6, 12, 18, 24
c) f = {factors of 18}
1, 2, 3, 6, 9, 18
d) t = {10 = 5x | 1 ≤ x ≤ 4, x ∈ n}
e) w = {prime numbers between 30 and 40}
31, 37

1. describe the following sets in words.

a) a = {8, 12, 16, 24}
b) b = {4, 9, 16, 25, 36}

1. riley owns a pizza shop. customers may choose any of the following toppings for their pizza:

• pepperoni
• green peppers
• pineapple
• bacon
• mushrooms
• onions
• ham
• sausage

a) let t = {pizza toppings available at rileys shop}. determine the value of n(t) and explain its meaning.
b) list the elements of m = {meat toppings available at rileys shop}. explain why m ⊂ t.
c) determine whether each of the following is a subset of t.
(i) {mushrooms, onions, pineapple}
(ii) {pepperoni, ham, olives, bacon}
(iii) the empty set
(iv) t

Explanation:

Part 1: List the elements of each set

Step1: Identify primes <25

Primes are numbers >1 with no divisors other than 1 and themselves.
$P = \{2, 3, 5, 7, 11, 13, 17, 19, 23\}$

Step2: List multiples of 3 <25

Multiply 3 by positive integers until result <25.
$M = \{3, 6, 9, 12, 15, 18, 21, 24\}$

Step3: Find factors of 18

List numbers that divide 18 evenly.
$F = \{1, 2, 3, 6, 9, 18\}$

Step4: Calculate set T

Substitute $x=1,2,3,4$ into $10-5x$.
For $x=1$: $10-5(1)=5$; $x=2$: $10-5(2)=0$; $x=3$: $10-5(3)=-5$; $x=4$: $10-5(4)=-10$
$T = \{-10, -5, 0, 5\}$

Step5: Identify primes between 30-40

Primes in this range have no divisors other than 1 and themselves.
$W = \{31, 37\}$

a) Identify the pattern: elements are $8 \times 1, 8 \times 2, 8 \times 3, 8 \times 4$.
b) Identify the pattern: elements are $4 \times 2, 4 \times 4, 4 \times 6, 4 \times 8$.

Step1: Find $n(T)$

Count total unique toppings in Riley's shop: Pepperoni, Mushrooms, Green Peppers, Onions, Pineapple, Ham, Bacon, Sausage.
$n(T) = 8$

Step2: List meat toppings M

Identify toppings classified as meat: Pepperoni, Ham, Bacon, Sausage. All are in set T.
$M = \{\text{Pepperoni}, \text{Ham}, \text{Bacon}, \text{Sausage}\}$

Step3: Check subsets of T

A subset only contains elements from the original set T.
i) All elements are in T; ii) Olives are not in T; iii) Empty set is a subset of every set; iv) A set is a subset of itself.

Answer:

a) $\{2, 3, 5, 7, 11, 13, 17, 19, 23\}$
b) $\{3, 6, 9, 12, 15, 18, 21, 24\}$
c) $\{1, 2, 3, 6, 9, 18\}$
d) $\{-10, -5, 0, 5\}$
e) $\{31, 37\}$

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Part 2: Describe sets in words