GE3-MMW

LECTURE: # 4 : RELATIONS AND FUNCTIONS:

Tasks:
1. Print this lecture for your individual copy
2. Copy and answer seatworks nos. 4, 8,20,21,24 abc, 25,26,27,28,29,30,31

1. A relation between two variables x and y is a set of ordered pairs

2. An ordered pair consist of a x and y-coordinate
3. A relation may be viewed as ordered pairs, mapping design, table, equation, or written in sentences
x-values are inputs, domain, independent variable
y-values are outputs, range, dependent variable
4. A relation is a rule that pairs each element in one set, called the domain, with one or more elements from the
second set called the range. It creates a set of ordered pairs. Example
Given:
Regular Holidays in the Philippines Month and Date
1. New Year’s day January 1
2. Labor Day May 1
3. Independence Day June 12
4. Bonifacio Day November 30
5. Rizal Day December 30

5. A clearer way to express a relation is to form a set of ordered pairs; These sets describe a relation
(New Year’s Day, January 1)
(Labor Day, May 1)
(Independence Day, June 12)
(Bonifacio Day, November 30)30)
(Rizal Day, December
6. {New Year’s day, Labor day, Independence Day, Bonifacio Day, Rizal Day, } are the domain of the relation.
7. {January 1, May 1, June 12, November 30, December 30 } the range of the relation.
8. TRUE OR FALSE:
1. {2,3}, {4,5} is a relation __________
2. {1,4}, {2,5}, {3,6 } is a relation ___________
9. Function – is a rule that pair each element in one set, called the domain, with exactly one element from a
second set, called the range.
10. This means that for each first coordinate, there is exactly one second coordinate or for every first element of
x , there corresponds a unique second element y.
11. Remember: A one-to-one correspondence and many-to-one correspondence are called Functions while one-
to-many correspondence is not
12. a function is…a relation in which every input is paired with exactly one output”
13. Is a relation a function?
14. Focus on the x-coordinates, when given a relation
15. If the set of ordered pairs have different x-coordinates,
it IS A function
16. If the set of ordered pairs have same x-coordinates,
it is NOT a function
17. Y-coordinates have no bearing in determining functions

18. The Functions can be represented using the following

a. Table
The perimeter of a square is four times the length of its side.
Side (S) 1 3 5 7 9
Perimeter (P) 4 12 20 28 36

b. Ordered pairs
{(1, 4), (3, 12), (5, 20), (7, 28), (9, 36)}

c. Mapping

1 4
 3 12
5 20
7 28
9 36
P2

d. Graphing- using vertical line Test, that is , a set of points in the plane is a graph of a function if and only
if no vertical line intersects the graph in more than one point.
The graph of a function f is the set of all points in the plane of the form (x, f(x)). We could also define
the graph of f to be the graph of the equation y = f(x). So, the graph of a function if a special case
of the graph of an equation.
19. Vertical Line Test: a relation is a function if a vertical line drawn through its graph, passes through only one
point.
AKA: “The Pencil Test”
Take a pencil and move it from left to right (–x to x); if it crosses more than one point, it is not a function

Given:

20.
{(0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0
What is the domain? {________________________}
21. What is the range? {________________________}
P3

22. Binary Functions – on a set is a calculation involving two elements of the set to produce another element of
the set.
23. A new math (binary ) operation, using the symbol *, is defined to be
a * b = 3a + b, where a and b are real numbers
24. Examples:
a. What is 4 * 3?
Solution: _______________________________
b. Is a * b commutative? That is is a * b = b * a ?
Verify: 3a + b = 3b + a
If a = 4 and b = 2,
Then, ____________________________________
____________________________________
c. Is a * b * c associative?
Solution: Verify if a * (b * c) = (a* b) * c
a * (3b + c) = (3a + b) + c
3a + (3b + c) = 3(3a + b) + c
if a = 2; b = 3; c = 4;
________________________________
____________________________________

{(0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0)}
25. Is this a function? ______________________
b. Hint: Look only at the x-coordinates
26.

{(–1, 7), (1, 0), (2, 3), (0, 8), (0, 5), (–2, 1)}
• Is this a function? _______________
• Hint: Look only at the x-coordinates
27. Which mapping represents a function? Choice 1 or choice 2?

3 –1
1 2
0 3
Choice 1
choice 2
Vertical Line Test
• Vertical Line Test: a relation is a function if a vertical line drawn through its graph,
passes through only one point.

AKA: “The Pencil Test”

Take a pencil and move it from left to right (–x to x); if it crosses more than one point,
it is not a function
P4

28. would this graph be a function? ____________

29. Is the following function discrete or continuous? What is the Domain? What is the Range?

30.

31.
For nos.29
TYPE: Discreet or Continuous ( example: graph 1 type is Discreet)
Domain: are all the points along x-axis ( Example: (-7, 1, 5,7,8,10)
Range: are all the coordinate points of the domain along y-axis (Example: (1,0,-7,5,2,8)
Elementary Logic:
31. A mathematical statement is logically true if it is true when the symbols in it are given their
standard mathematical meaning
32. A logic consists of a set of statements (syntax), an assignment of meaning to the statements
(semantics), and a method of proving statements.
33. A statement in logic L is valid if it is true in all interpretations of L.
34. Three part statements called categorical syllogisms. Example of a categorical syllogism:
All P are Q. All Q are R. Thus all P are R.
35. An interpretation of this syllogism:
All ducks are sponges. All sponges are happy. Therefore all ducks are happy.
36. The syllogism is true if one of the hypotheses is false or the conclusion is true
37. Terms – basic elements that make up a language system.
38. Logic – science of correct reasoning.!
39. A proposition for statement is a sentence that is either true or false (without additional information)
40. Reasoning – The drawing of inferences or conclusions from known or assumed facts
41. Deductive Reasoning – A type of logic in which one goes from a general statement to a specific
instance.
42. Syllogism: An argument composed of two statements or premises (the major and minor premises),
followed by a conclusion.
43. an example of a syllogism.
All men are mortal. (major premise)
Socrates is a man. (minor premise)
Therefore, Socrates is mortal. (conclusion)
44. The above is For any given set of premises, if the conclusion is guaranteed, the arguments is said
to be valid.
45. If the conclusion is not guaranteed (at least one instance in which the conclusion does not follow),
the argument is said to be invalid.
46. Logic terms:
1. Inference – the process of deducing or extracting a statement (conclusion) from the
previous statement/s.
2. Argument – the verbal expression of inference.
3. Syllogism – the format of arguments with three statements.
4. Conclusion – the statement being supported.
5. Premises – the statement/s that support/s the conclusion.