Given \(A = \{1,2,3,4\}\), define relations on \(A\) having the properties:
Let \(R\) be a relation defined on the set of natural numbers \(\mathbb{N}\) as follows:
\[ R = \{(x,y) : x,y \in \mathbb{N},\ 2x + y = 41 \}. \]
Find the domain and range of the relation \(R\). Also verify whether \(R\) is reflexive, symmetric and transitive.
Domain of \(R = \{1,2,3,4,\ldots,20\}\)
Range of \(R = \{1,3,5,7,9,\ldots,39\}\)
\(R\) is neither reflexive, nor symmetric, nor transitive.
Given \(A = \{2,3,4\}\) and \(B = \{2,5,6,7\}\), construct an example of each of the following:
Give an example of a map:
Let \(A = \mathbb{R} - \{3\}\) and \(B = \mathbb{R} - \{1\}\). Let \(f : A \to B\) be defined by
\[ f(x) = \dfrac{x - 2}{x - 3}. \]
Show that \(f\) is bijective.
Let \(A = [-1,1]\). Discuss whether the following functions defined on \(A\) are one-one, onto or bijective:
(i) \(f(x) = \dfrac{x}{2}\)
(ii) \(g(x)=|x|\)
(iii) \(h(x) = x|x|\)
(iv) \(k(x)=x^2\)
(i) \(f\) is one-one but not onto.
(ii) \(g\) is neither one-one nor onto.
(iii) \(h\) is bijective.
(iv) \(k\) is neither one-one nor onto.
Each of the following defines a relation on \(\mathbb{N}\):
(i) \(x\) is greater than \(y\)
(ii) \(x + y = 10\)
Determine which of the above relations are reflexive, symmetric and transitive.
(i) Transitive
(ii) Symmetric
(iii) Reflexive, symmetric and transitive
(iv) Transitive
Let \(A = \{1,2,3,\ldots,9\}\) and \(R\) be the relation defined on \(A \times A\) by
\[(a,b) R (c,d) \text{ if } a+d = b+c.\]
Prove that \(R\) is an equivalence relation and find the equivalence class \([(2,5)]\).
\([(2,5)] = \{(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)\}
Using the definition, prove that the function \(f : A \to B\) is invertible if and only if \(f\) is both one-one and onto.
Functions \(f,g : \mathbb{R} \to \mathbb{R}\) are defined by \(f(x)=x^2+3x+1\) and \(g(x)=2x−3\). Find:
(i) \(f \circ g\)
(ii) \(g \circ f\)
(iii) \(f \circ f\)
(iv) \(g \circ g\)
(i) \((f \circ g)(x) = 4x^2 - 6x + 1\)
(ii) \((g \circ f)(x) = 2x^2 + 6x - 1\)
(iii) \((f \circ f)(x)= x^4 + 6x^3 + 14x^2 + 15x + 5\)
(iv) \((g \circ g)(x)=4x - 9\)
Let * be the binary operation defined on \(\mathbb{Q}\). Determine which of the following operations are commutative:
(i) \(a * b = a - b\)
(ii) \(a * b = a^2 + b^2\)
(iii) \(a * b = a + ab\)
(iv) \(a * b = (a - b)^2\)
(ii) & (iv)
Let * be a binary operation defined on \(\mathbb{R}\) by \(a * b = 1 + ab\). Then determine whether the operation is:
(i) commutative but not associative
(ii) associative but not commutative
(iii) neither commutative nor associative
(iv) both commutative and associative
(i)