Finding the Intersection of Sets
Find the intersection of the solution sets of the following linear equations: \(2x - 3y = 6\) and \(5x + 4y = 20\).
Finding the Union of Number Sets
Let \( S = \{1, 2, 3\} \) and \( T = \{3, 4, 5\} \) be two sets of numbers. What is the union of these two sets?
Determining if a Set is a Subset of Another Set
Let \(A = \{1, 2, 3\}\) and \(B = \{1, 2, 3, 4, 5\}\) be two sets of numbers. Is set \(A\) a subset of set \(B\)?
Finding the Set Complement of Two Sets
Let \( A = \{1, 2, 3, 4, 5\} \) and \( B = \{3, 4, 5, 6, 7\} \) be two sets in the universal set \( U = \{1, 2, 3, 4, 5, 6, 7\} \). Find the complement of the set \( A \bigcup B \) in \( U \).
Finding the Power Set
Let's consider the set \( S = \{1, 2\} \). What is the power set of \( S \)?
Finding the Cardinality
Let's say we have a set of linear vectors \(V\) in a 3-dimensional space, where \(V = \{v_1, v_2, v_3, v_4\}\). The vectors are defined as follows: \(v_1 = (1, 0, 0)\), \(v_2 = (0, 1, 0)\), \(v_3 = (0, 0, 1)\) and \(v_4 = (1, 1, 1)\). What is the cardinality of the base of vector space that these vectors span?
Finding the Cartesian Product of Two Sets
Let \(A = \{1, 2\}\) and \(B = \{3, 4\}\). Find the Cartesian product of these two sets (\(A \times B\)).
Determining if a Set is a Proper Subset of Another Set
Let's consider two sets in the linear algebra context. Set A contains the vectors [1, 2], [3, 4] and set B contains the vectors [1, 2], [3, 4], [5, 6]. Is set A a proper subset of set B?