If \(\left(\dfrac{x}{3} + 1,\; y - \dfrac{2}{3}\right) = \left(\dfrac{5}{3},\; \dfrac{1}{3}\right)\), find the values of \(x\) and \(y\).
\(x = 2\) and \(y = 1\).
If the set \(A\) has 3 elements and the set \(B = \{3, 4, 5\}\), then find the number of elements in \(A \times B\).
The number of elements in \(A \times B\) is \(9\).
If \(G = \{7, 8\}\) and \(H = \{5, 4, 2\}\), find \(G \times H\) and \(H \times G\).
\(G \times H = \{(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)\}\)
\(H \times G = \{(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)\}\)
State whether each of the following statements is true or false. If the statement is false, rewrite it correctly.
(i) If \(P = \{m, n\}\) and \(Q = \{n, m\}\), then \(P \times Q = \{(m, n), (n, m)\}\).
(ii) If \(A\) and \(B\) are non-empty sets, then \(A \times B\) is a non-empty set of ordered pairs \((x, y)\) such that \(x \in A\) and \(y \in B\).
(iii) If \(A = \{1, 2\}\), \(B = \{3, 4\}\), then \(A \times (B \cap \varnothing) = \varnothing\).
(i) False. \(P \times Q = \{(m, n), (m, m), (n, n), (n, m)\}\).
(ii) True.
(iii) True.
If \(A = \{-1, 1\}\), find \(A \times A\) and \(A \times A \times A\).
\(A \times A = \{(-1, -1), (-1, 1), (1, -1), (1, 1)\}\)
\(A \times A \times A = \{(-1, -1, -1), (-1, -1, 1), (-1, 1, -1), (-1, 1, 1), (1, -1, -1), (1, -1, 1), (1, 1, -1), (1, 1, 1)\}\)
If \(A \times B = \{(a, x), (a, y), (b, x), (b, y)\}\), find the sets \(A\) and \(B\).
\(A = \{a, b\}\), \(B = \{x, y\}\).
Let \(A = \{1, 2\}\), \(B = \{1, 2, 3, 4\}\), \(C = \{5, 6\}\) and \(D = \{5, 6, 7, 8\}\). Verify that
(i) \(A \times (B \cap C) = (A \times B) \cap (A \times C)\)
(ii) \(A \times C\) is a subset of \(B \times D\).
Let \(A = \{1, 2\}\) and \(B = \{3, 4\}\). Write \(A \times B\). How many subsets will \(A \times B\) have? List them.
\(A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4)\}.\)
\(A \times B\) will have \(2^{4} = 16\) subsets.
Let \(A\) and \(B\) be two sets such that \(n(A) = 3\) and \(n(B) = 2\). If \((x, 1), (y, 2), (z, 1)\) are in \(A \times B\), find \(A\) and \(B\), where \(x, y\) and \(z\) are distinct elements.
\(A = \{x, y, z\}\) and \(B = \{1, 2\}\).
The Cartesian product \(A \times A\) has 9 elements among which are found \((-1, 0)\) and \((0, 1)\). Find the set \(A\) and the remaining elements of \(A \times A\).
\(A = \{-1, 0, 1\}.\)
The remaining elements of \(A \times A\) are \((-1, -1), (-1, 1), (0, -1), (0, 0), (1, -1), (1, 0), (1, 1)\).