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 \((-1, -1), (-1, 1), (0, -1), (0, 0), (1, -1), (1, 0), (1, 1)\).
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 \(R = \mathbb{Z}\)
Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.
(i) {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)}
(ii) {(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)}
(iii) {(1,3), (1,5), (2,5)}
(i) Yes, Domain = \(\{2, 5, 8, 11, 14, 17\}\), Range = \(\{1\}\)
(ii) Yes, Domain = \(\{2, 4, 6, 8, 10, 12, 14\}\), Range = \(\{1, 2, 3, 4, 5, 6, 7\}\)
(iii) No.
Find the domain and range of the following real functions:
(i) \(f(x) = -|x|\)
(ii) \(f(x) = \sqrt{9 - x^2}\)
(i) Domain = \(\mathbb{R}\), Range = \(( -\infty, 0 ]\)
(ii) Domain = \(\{x : -3 \le x \le 3\}\), Range = \(\{x : 0 \le x \le 3\}\)
A function \(f\) is defined by \(f(x) = 2x - 5\). Write the values of:
(i) \(f(0)\)
(ii) \(f(7)\)
(iii) \(f(-3)\)
(i) \(f(0) = -5\)
(ii) \(f(7) = 9\)
(iii) \(f(-3) = -11\)
The function \(t\) which maps temperature in degree Celsius into degree Fahrenheit is defined by
\[ t(C) = \dfrac{9C}{5} + 32. \]
Find:
(i) \(t(0)\)
(ii) \(t(28)\)
(iii) \(t(-10)\)
(iv) The value of \(C\) when \(t(C)=212\)
(i) 32
(ii) \(\dfrac{412}{5}\)
(iii) 14
(iv) 100
Find the range of each of the following functions.
(i) \(f(x) = 2 - 3x, \ x \in \mathbb{R}, x > 0\)
(ii) \(f(x) = x^2 + 2, x\) is a real number
(iii) \(f(x) = x, x\) is a real number
(i) Range = \(( -\infty, 2 )\)
(ii) Range = \([2, \infty)\)
(iii) Range = \(\mathbb{R}\)