What is a set in mathematics?

A set is a collection of distinct objects, considered as an object in its own right. Sets are usually denoted by curly braces. For example, the set of even numbers can be written as {2, 4, 6, 8, ...}.

What is the union of two sets?

The union of two sets A and B, denoted as A ∪ B, is the set of elements that are in A, in B, or in both. For example, if A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.

What does it mean for a set to be a subset?

A set A is a subset of a set B, denoted A ⊆ B, if every element of A is also an element of B. For example, if A = {1, 2} and B = {1, 2, 3}, then A is a subset of B.

How do you find the intersection of two sets?

The intersection of two sets A and B, denoted as A ∩ B, is the set of elements that are common to both A and B. For example, if A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}.

Given the sets A = {1, 2, 3} and B = {3, 4, 5}, what is A ∪ B and A ∩ B?

The union A ∪ B = {1, 2, 3, 4, 5} includes all unique elements from both sets. The intersection A ∩ B = {3} includes only the elements that are in both sets.

If A = {x | x is an even integer} and B = {x | x is a positive integer}, what is A ∩ B?

A ∩ B = {2, 4, 6, 8, ...} includes all positive even integers. This set consists of elements common to both sets.

What is the formula to find the number of subsets of a set with n elements?

The number of subsets of a set with n elements is given by 2ⁿ. For example, if n = 3, the set has 2³ = 8 subsets.

How many elements are in the power set of A = {1, 2, 3}?

The power set of A contains all possible subsets of A, including the empty set and A itself. Since A has 3 elements, the power set has 2³ = 8 elements.

If A = {1, 2, 3} and B = {3, 4, 5}, what is the number of elements in A ∪ B?

A ∪ B = {1, 2, 3, 4, 5} has 5 elements. Therefore, the number of elements in A ∪ B is 5.