Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

Question

Which sets of ordered pairs represent functions from \(A\) to \(B\) ? Explain. \(A=\\{a, b, c\\}\) and \(B=\\{0,1,2,3\\}\) (a) \(\\{(a, 1),(c, 2),(c, 3),(b, 3)\\}\) (b) \(\\{(a, 1),(b, 2),(c, 3)\\}\) (c) \(\\{(1, a),(0, a),(2, c),(3, b)\\}\)

Step-by-Step Solution

Answer

Only set (b) represents a function from set 'A' to set 'B'.

Step 1: Analyzing set (a)

Set \(a\) is \(\{(a, 1),(c, 2),(c, 3),(b, 3)\}\). Here, the element 'c' from set 'A' corresponds to two elements '2' and '3' in set 'B'. Therefore, set \(a\) does not represent a function.

Step 2: Analyzing set (b)

Set \(b\) is \(\{(a, 1),(b, 2),(c, 3)\}\). In this set, 'a', 'b' and 'c' all correspond to one element each in set 'B' hence, set \(b\) represents a function from 'A' to 'B'.

Step 3: Analyzing set (c)

Set \(c\) is \(\{(1, a),(0, a),(2, c),(3, b)\}\). Here, the elements of set 'B' are paired with elements from set 'A'. Since the question specifically asks for functions from 'A' to 'B', set \(c\) does not represent such a function.

Key Concepts

Ordered Pairs

Ordered pairs are fundamental in understanding relations between two sets. In mathematics, an ordered pair consists of two elements arranged in a specific sequence. This sequence is critical, as it signifies the relationship between elements of different sets. Generally, an ordered pair can be represented as \((x, y)\), where "\(x\)" is an element from the first set, and "\(y\)" is from the second set.

- For example, in the ordered pair \((a, 1)\), "\(a\)" is from set "\(A\)", and "\(1\)" is from set "\(B\)". - The order matters because \((a, 1)\) is not the same as \((1, a)\).
In the context of functions, each element from the domain (first set) must pair with exactly one element from the range (second set). This helps us determine whether a set of ordered pairs is a function or not.

Domain and Range

Domain and range are two key concepts in understanding functions. They help distinguish between valid and invalid function pairs.The domain of a function is the set of all possible inputs. That is, the elements from which the function can take an initial value. In the problem, the domain is set \(A = \{a, b, c\}\).

The range, on the other hand, is the set of all possible outputs. These are the elements that are paired with the domain through the function. In this case, set \(B = \{0, 1, 2, 3\}\) acts as the range.
- A function requires each element in the domain to pair with only one element in the range. - Hence, if an element from \(A\) pairs with more than one element in \(B\), it invalidates the set from being a function.

Function Analysis

Analyzing whether a set of ordered pairs represents a function involves checking the pairings between the domain and range.To determine if a set of ordered pairs is a function from one set to another:

• Each element in the domain must be paired with exactly one element in the range.
• If any element from the domain is paired with more than one distinct element in the range, it is not a function.

In this scenario, let's evaluate the sets:

- In set (a), 'c' pairs with both '2' and '3', which means it does not fulfill the function's requirement.- Set (b) fulfills the function criteria as each element from \(A\) pairs with a unique element in \(B\).- Set (c) does not follow the specified direction from \(A\) to \(B\), indicating it is not a function as per the question's conditions.Function analysis is crucial in mathematics as it helps in mapping inputs to desired outputs in a well-defined manner.