Question

let u = {1,2,3,4,5,6,7,8,9,10} be the universal set and a = {2,4,6,8,10}. what is the complement of a, denoted a^c? {1,3,5,7,9} {2,4,6,8,10} u ∅ multiple choice 1 point for any sets a and b, the union operation is commutative, meaning a ∪ b = b ∪ a. true false multiple choice 1 point the symmetric difference of two sets a and b, denoted aδb, can be expressed as (a \\ b) ∪ (b \\ a). true false multiple choice 1 point for any sets a, b, and c, the expression a ∩ (b ∪ c) is equivalent to (a ∩ b) ∪ (a ∩ c). true false

Explanation:

Step1: Recall complement definition

The complement of a set \(A\) in the universal set \(U\), denoted \(A^{c}\), is the set of all elements in \(U\) that are not in \(A\).

Step2: Identify elements in \(U\) not in \(A\)

Given \(U=\{1,2,3,4,5,6,7,8,9,10\}\) and \(A = \{2,4,6,8,10\}\), then \(A^{c}=\{1,3,5,7,9\}\).

Step3: Recall union - commutativity

The union of two sets \(A\) and \(B\), \(A\cup B\), is the set of all elements that are in \(A\) or in \(B\). By definition, \(A\cup B=B\cup A\), so the union operation is commutative.

Step4: Recall symmetric - difference definition

The symmetric difference \(A\Delta B\) of two sets \(A\) and \(B\) is defined as the set of elements which are in either of the sets \(A\) and \(B\) but not in their intersection. And \(A\Delta B=(A - B)\cup(B - A)\) (where \(A - B=A\setminus B\) is the set of elements in \(A\) but not in \(B\)).

Step5: Recall distributive law of sets

The distributive law of intersection over union states that for any sets \(A\), \(B\), and \(C\), \(A\cap(B\cup C)=(A\cap B)\cup(A\cap C)\).

Answer:

1. A. \(\{1,3,5,7,9\}\)
2. A. True
3. A. True
4. A. True