Example Question - ordered pairs
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Set Theory and Relations Worksheet
This image appears to show a question from an academic worksheet or test, related to set theory and relations. (a) The first part of the question asks you to state the (i) object part 3, (ii) the relation in the form of a set of ordered pairs. From the diagram, we can see that Set A includes the elements {1, 2, 3, 4} and Set B includes the elements {5, 6, 7, 8}. (i) The object part 3 in Set A appears to be related to the number 7 in Set B. (ii) As for the relation in the form of a set of ordered pairs, we would read the arrows connecting elements of Set A to elements of Set B to form these pairs. The complete set of ordered pairs, assuming we can see all links between the sets, appears to be: R = {(1, 5), (1, 6), (2, 6), (2, 7), (3, 7), (4, 8)} (b) For the second part of the question, we need to determine whether this relation is a function or not and provide reasoning for the answer. A relation from Set A to Set B is a function if and only if every element of Set A is related to no more than one element in Set B. Looking at the ordered pairs and the diagram: 1 is related to 5 and 6, which is more than one element of Set B. 2 is related to 6 and 7, which is more than one element of Set B. 3 is related only to 7. 4 is related only to 8. Since at least one element of Set A (specifically 1 and 2) is related to more than one element of Set B, the relation is NOT a function. Thus, your answer to part (b) would be that the relation is not a function because there exists an element in Set A that is related to more than one element in Set B.
Graphing a Linear Equation
To solve the equation and complete the ordered pairs, you need to plug the x-values from each ordered pair into the equation y = -x/5 + 2 and solve for y. The equation is linear, which means you will be graphing a straight line. You only need two points to define a straight line, but let's find the y-values for all the given x-values: 1. For the ordered pair (-5, _), plug in x = -5: y = -(-5)/5 + 2 y = 5/5 + 2 y = 1 + 2 y = 3 So, the complete ordered pair is (-5, 3). 2. For the ordered pair (0, _), plug in x = 0: y = -(0)/5 + 2 y = 0 + 2 y = 2 So, the complete ordered pair is (0, 2). 3. For the ordered pair (5, _), plug in x = 5: y = -(5)/5 + 2 y = -1 + 2 y = 1 So, the complete ordered pair is (5, 1). To graph these points, you just need to put these coordinates on a graph and draw a straight line through them. Using (-5, 3) and (5, 1), you can draw the graph as follows: - Plot the point (-5, 3) on the graph. - Plot the point (5, 1) on the graph. - Draw a straight line that passes through both points, extending it on both ends. This line is the graphical representation of the solution set for the equation y = -x/5 + 2.
Identifying Non-Functional Sets of Ordered Pairs
The question asks which set of ordered pairs does not represent a function. For a set of ordered pairs to represent a function, each input (the first number in each pair, also known as the x-value) must have exactly one output (the second number in each pair, also known as the y-value). If an input is paired with more than one output, the set of pairs does not represent a function. Looking at the options: A) (1,0), (3,18), (5,26), (7,34), (9,42) - Each input has a unique output, thus represents a function. B) (2,10), (3,24), (4,15), (5,6), (6,25) - Each input has a unique output, thus represents a function. C) (0,8), (5,4), (10,0), (15,4), (20,8) - Each input has a unique output, thus represents a function. D) (1,9), (2,6), (2,3), (6,4), (9,5) - The input '2' is paired with two different outputs '6' and '3'. This means the set does not represent a function. The correct answer is D, because it has a repeated x-value with different y-values.
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