Let \(A = \{1, 2, 3, \ldots, 14\}\). Define a relation \(R\) from \(A\) to \(A\) by \(R = \{(x, y) : 3x - y = 0,\; x, y \in A\}\). Write down its domain, codomain and range.
\(R = \{(1, 3), (2, 6), (3, 9), (4, 12)\}\)
Domain of \(R = \{1, 2, 3, 4\}\)
Range of \(R = \{3, 6, 9, 12\}\)
Codomain of \(R = A = \{1, 2, \ldots, 14\}\)
Define a relation \(R\) on the set \(\mathbb{N}\) of natural numbers by \(R = \{(x, y) : y = x + 5,\; x\) is a natural number less than \(4;\; x, y \in \mathbb{N}\}\). Depict this relation in roster form. Write down the domain and the range.
\(R = \{(1, 6), (2, 7), (3, 8)\}\)
Domain of \(R = \{1, 2, 3\}\)
Range of \(R = \{6, 7, 8\}\)
Let \(A = \{1, 2, 3, 5\}\) and \(B = \{4, 6, 9\}\). Define a relation \(R\) from \(A\) to \(B\) by \(R = \{(x, y) :\) the difference between \(x\) and \(y\) is odd, \(x \in A, y \in B\}\). Write \(R\) in roster form.
\(R = \{(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)\}\)
The figure (Fig. 2.7) shows a relationship between the sets \(P\) and \(Q\). Write this relation
(i) in set-builder form (ii) in roster form. What is its domain and range?
(i) \(R = \{(x, y) : y = x - 2 \text{ for } x = 5, 6, 7\}\)
(ii) \(R = \{(5, 3), (6, 4), (7, 5)\}\)
Domain of \(R = \{5, 6, 7\}\)
Range of \(R = \{3, 4, 5\}\)
Let \(A = \{1, 2, 3, 4, 6\}\). Let \(R\) be the relation on \(A\) defined by \(R = \{(a, b) : a, b \in A, b \text{ is exactly divisible by } a\}\).
(i) Write \(R\) in roster form.
(ii) Find the domain of \(R\).
(iii) Find the range of \(R\).
(i) \(R = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 4), (2, 6), (2, 2), (4, 4), (6, 6), (3, 3), (3, 6)\}\)
(ii) Domain of \(R = \{1, 2, 3, 4, 6\}\)
(iii) Range of \(R = \{1, 2, 3, 4, 6\}\)
Determine the domain and range of the relation \(R\) defined by \(R = \{(x, x + 5) : x \in \{0, 1, 2, 3, 4, 5\}\}\).
Domain of \(R = \{0, 1, 2, 3, 4, 5\}\}
Range of \(R = \{5, 6, 7, 8, 9, 10\}\)
Write the relation \(R = \{(x, x^3) : x\) is a prime number less than \(10\}\) in roster form.
\(R = \{(2, 8), (3, 27), (5, 125), (7, 343)\}\)
Let \(A = \{x, y, z\}\) and \(B = \{1, 2\}\). Find the number of relations from \(A\) to \(B\).
Number of relations from \(A\) into \(B\) is \(2^6\).
Let \(R\) be the relation on \(\mathbb{Z}\) defined by \(R = \{(a, b) : a, b \in \mathbb{Z}, a - b \text{ is an integer}\}\). Find the domain and range of \(R\).
Domain of \(R = \mathbb{Z}\)
Range of \(R = \mathbb{Z}\)