Set Theory and Functions Quiz

Set Theory and Subsets

• The number of subsets of a set with five elements is 32. This is because each element can be either included or excluded from a subset, so there are 2 possibilities for each element. With 5 elements, there are 2^5 = 32 total possible subsets.
• If |A| = 50, |A ∩ B| = 44, and |B| = 58, then |P(A - B)| = 2^6. We use the formula |P(A - B)| = 2^(|A - B|), and to find |A - B| we utilize |A - B| = |A| - |A ∩ B| which equals 6.

Cardinality and Sets

• In a survey of 1000 consumers, if 72 liked product A and 45 liked product B, the least number that must have liked both products is 17. This is found by utilizing the principle of inclusion-exclusion: |A ∪ B| = |A| + |B| - |A ∩ B|. Since we are looking for the least number that liked both products, we assume the maximum overlap, making |A ∪ B| = 1000. Therefore, |A ∩ B| = 72 + 45 - 1000 = 17.

Ordered Pairs and Functions

• If (x^2 - 4x + 5, y^2 - 63) = (x^2 + 1, 1), then x=2 and y = ±8. This comes from equating each part of the ordered pair: x^2 - 4x + 5 = x^2 + 1 and y^2 - 63 = 1. Solving the first equation leads to x = 2, and solving the second equation gives us y = ±8.

Function Types

• A function f: R → R defined by f(x) = x, ∀x∈R is: (i) Injective: Yes, each element in the range has exactly one preimage. (ii) Surjective: Yes, every element in the range is mapped by the function. (iii) Bijective: Yes, since it's both injective and surjective. (iv) Not defined: No, it is a well-defined function.

Domain and Range of Functions

• For functions from R to R, we need to consider cases where the function becomes undefined (e.g., dividing by zero or taking the square root of a negative number). Therefore,
  • The domain of the function f(x) = √(9 - x^2) is x ∈ [-3, 3], as we need 9 - x^2 ≥ 0.
  • The domain of the function f(x) = √(x - 5) is x ∈ [5, ∞), as we need x - 5 ≥ 0.
  • The domain of the function f(x) = √(16 - x^2) is x ∈ [-4, 4], as we need 16 - x^2 ≥ 0.
  • The domain of the function f(x) = x^2 - x - 110 is x ∈ R (the set of all real numbers) since it's defined for all x values.
  • The domain of the function f(x) = cos(x) - 3 is x ∈ R (the set of all real numbers) as the cosine function is defined for all x values.
  • Remember to find the range, consider the behavior and limitations of each function.

Maximum, Minimum, Supremum, Infimum, and Boundedness

• Given a set of numbers, we can determine its maximum, minimum, supremum, and infimum, which refer to the "largest", "smallest", "least upper bound", and "greatest lower bound" respectively.
• Considering boundedness, we can classify sets as bounded above (having an upper bound), bounded below (having a lower bound), or bounded (having both an upper and lower bound).
• Here are examples of the concepts applied to the given sets:*

Set Analysis

• S1 = {n/n : n ∈ N} {1} maximum, minimum, supremum, and infimum are all 1, and the set is bounded
• S3 = {n(-1)^n/n : n ∈ N} {1} maximum, minimum, supremum, and infimum are all 1, and the set is bounded
• S5 = {x ∈ R : x < 1} {1} maximum, minimum, supremum, and infimum are all 1, and the set is bounded above
• S6 = {x ∈ R : x^2 - 3x + 2 < 0} {1} maximum, minimum, supremum, and infimum are all 1, and the set is bounded
• S7 = {x ∈ R : x^2 - 3x + 2 > 0} {1} maximum, minimum, supremum, and infimum are all 1, and the set is bounded
• S8 = {x ∈ Q : x^2 - 2 ≤ 0}, where Q is the set of rational numbers {√2} maximum, minimum, supremum, and infimum are all √2, and the set is bounded