Exploring Sets and Notations

What is the set builder notation for the set of all integers that can be expressed as the sum of an even integer and a multiple of 4?

• { x | x = n^2, n ∈ ℤ }

• { x | x = 2n + 3, n ∈ ℤ }

• { x | x = 2n, n ∈ ℤ }

• { x | x = 4m + 2, m ∈ ℤ }

Determine whether the set of all even integers less than 10, represented as {0, 2, 4, 6, 8}, is finite or infinite.

• undefined

• countable

• finite

• infinite

Determine the cardinality of the set {a, b, c, d, e, f} and explain the significance of this value in the context of set theory.

• 5

• 6

• 7

• 8

Convert the set {x | x is a prime number less than 20} to roster notation.

• {1, 4, 6, 8, 9}

• {2, 3, 5, 7, 11, 13, 17, 19}

• {2, 3, 5, 9, 15}

• {2, 3, 5, 7, 11, 13, 17}

Considering the properties of the set of all natural numbers, can you determine whether it is finite or infinite, and provide a justification for your answer?

• infinite

• finite

• a set of numbers less than 100

• a set of even numbers only

What is the cardinality of the set of all even integers between 1 and 20 inclusive?

• 10

• 5

• 20

• 15

Express the set of all positive integers in set builder notation, ensuring to include a condition that distinguishes them from non-integers.

• {x | x ∈ Z, x ≤ 0}

• {x | x ∈ Z, x < 1}

• {x | x ∈ R, x > 0}

• {x | x ∈ Z, x > 0}

Convert the roster notation {2, 4, 6, 8, 10, 12, 14} to set builder notation, ensuring to include all even integers up to 14.

• {x | x is an even integer and 2 ≤ x ≤ 14}

• {x | x is an even integer and 1 ≤ x ≤ 13}

• {x | x is an odd integer and 1 ≤ x ≤ 15}

• {x | x is an integer and 0 < x < 16}

Analyze the set of vowels in the English alphabet and determine its cardinality, considering both uppercase and lowercase representations.

• 10

• 3

• 5

• 7

Considering the properties of the set of all real numbers, can you determine whether it is finite, infinite, or possesses other characteristics such as being bounded or countable?

• infinite

• finite

• bounded

• countable