Relations and functions:
The chapter "Relations and functions" carrying the weightage of \(16\) marks in the board examination. It covers the concepts like Ordered pair, Cartesian product, Relations, functions, Representation and types of functions, and Composition of functions.
| Total marks \(16\) |
Total marks \(= 16\)
|
To prepare well for the board examination, it is necessary to understand the following concepts clearly.
- Cartesian product - Find the sets.
- Relations - Determine domain, range, codomain and write the relation using set builder form and Roster form.
- Functions - Verify the relation is a function, find the value of x.
- Representation of function - Set of ordered pairs, table form, Arrow diagram and Graph.
- Types of function - One - one function, Many - one function, Onto function, Into function, Bijection.
Important!
It is highly recommended to learn and understand how to solve all the MCQ's available in textbook at the end of each chapter and just before the unit exercise section.
| Important Concept (Learning Outcomes) | Expected Question Type | Concept dealt with |
| Ordered Pairs and Cartesian Products | Sec A, Sec B | |
| Verification of set properties related to Cartesian product | Sec C | Verify the relation |
| Composition of Function | Sec B | Composition of function |
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Determining if a given relation is a function and finding its domain, co-domain, and range from an arrow diagram or a set of ordered pairs
Find the ordereded pairs, table, arrow diagram and graph
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Sec C |
Let us recall the concepts in Relations and functions:
1. If \(A\) and \(B\) are two non-empty sets, then the set of all ordered pairs \((a,b)\) such that \(a \in A\), \(b \in B\) is called the Cartesian product of \(A\) and \(B\), and is denoted by \(A \times B\).
2. Let us take any two non-empty sets as \(A\) and \(B\). A ‘relation’ \(R\) from \(A\) to \(B\) is a subset of \(A×B\) satisfying some specified conditions. If \(x ∈ A\) is related to \(y ∈ B\) through \(R\) , then we write it as \(x Ry\). \(x Ry\) if and only if \((x,y) ∈ R\).
3. The domain is the set used as an input in a function.
4. A co-domain is a set that includes all the possible values of a given function.
5. The range is the set of values that actually do come out. Range is the co-domain's subset. Remember, all ranges are co-domains but not all the co-domains are ranges.
6. A relation \(f\) between two non-empty sets \(X\) and \(Y\) is called a function from \(X\) to \(Y\) if, for each \(x \in X\), there exists only one \(y \in Y\) such that \((x, y) \in f\).
That is, \(f\) \(=\) \(\{(x, y)\)\(|\) for all \(x \in X\),\(y \in Y\).
That is, \(f\) \(=\) \(\{(x, y)\)\(|\) for all \(x \in X\),\(y \in Y\).
7. Vertical line test: A function is represented by a curve on a graph.
When a vertical line is drawn and intersects the curve precisely at one point, the curve forms a function.
8. If for all \(a_1, a_2 \in A\), \(f(a_1) = f(a_2)\) implies \(a_1 = a_2\), then \(f\) is a one-to-one function.
9. Let \(f: A \rightarrow B\) be a many-to-one function. Then, two or more elements in \(A\) will have a distinct image in \(B\).
10. A function \(f: A \rightarrow B\) is said to be onto function if the range of \(f\) is equal to the co-domain of \(f\).
11. A function \(f: A \rightarrow B\) is called an into function if there exists at least one element in \(B\) which is not the image of any element of \(A\).
12. A function that is both one-to one and onto is called a bijection.
13. Let \(f: A \rightarrow B\) and \(g: B \rightarrow C\) be two functions, then the composition of \(f\) and \(g\) denoted by \(g \circ f\) is defined as the function \(g \circ f(x) = g(f(x))\) for all \(x \in A\).