Second Examination For GE 02-mathematics In The Modern World Part I
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Questions and Answers
1.
Which relation is described as a one-to-one correspondence and many-to-one correspondence?
A.
Logic
B.
Set
C.
Binary function
D.
Function
Correct Answer D. Function
Explanation A function is described as a one-to-one correspondence when each element in the domain corresponds to exactly one element in the range. It is also described as a many-to-one correspondence when multiple elements in the domain correspond to the same element in the range. Therefore, a function can be described as both a one-to-one correspondence and a many-to-one correspondence.
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2.
What do you call the process of gathering specific information usually done through observation and measurement?
A.
Analogy
B.
Deductive reasoning
C.
Inductive reasoning
D.
Intuition
Correct Answer C. Inductive reasoning
Explanation Inductive reasoning is the process of gathering specific information through observation and measurement. It involves making generalizations or forming conclusions based on specific instances or examples. This method is commonly used in scientific research and data analysis, where patterns and trends are identified from specific observations to make broader predictions or theories.
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3.
Is a statement equivalent to the negation of a statement.
A.
TautologyÂ Â Â Â Â Â Â Â Â Â
B.
Denial
C.
Inverse
D.
Contrapositive
Correct Answer B. Denial
Explanation The term "denial" refers to the act of contradicting or refusing to accept a statement. In the context of this question, if a statement is equivalent to the negation of another statement, it means that the two statements contradict each other. Therefore, "denial" is an appropriate term to describe this relationship. "Tautology" refers to a statement that is always true, "inverse" refers to a statement formed by negating both the subject and predicate of the original statement, and "contrapositive" refers to a statement formed by negating and switching the subject and predicate of the original statement. None of these terms accurately describe the relationship between a statement and its negation.
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4.
It refers to a set that contains all the elements considered in a particular situation.
A.
Subset
B.
Equal set
C.
Universal set
D.
Proper subset
Correct Answer C. Universal set
Explanation The correct answer is "universal set". A universal set refers to a set that contains all the elements considered in a particular situation. It includes all possible elements that can be part of any subset or proper subset.
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5.
It is a sentence that is either true or false.
A.
Proposition
B.
Logic
C.
Predicate
D.
Contrapositive
Correct Answer A. Proposition
Explanation A proposition is a statement that can be either true or false. It is a fundamental concept in logic, where propositions are used to build logical arguments and reasoning. Unlike predicates, which involve variables and can be true or false depending on the values assigned to the variables, propositions are standalone statements that are evaluated as either true or false. The term "contrapositive" refers to a specific type of logical statement, but it is not directly related to the concept of propositions.
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6.
If 3 = (9/3) and (9/3) = (30/10) , then 3 = (30/10). What property of real numbers is being illustrated?
A.
Reflexive
B.
Symmetric
C.
Commutative
D.
Transitive
Correct Answer D. Transitive
Explanation The property of real numbers being illustrated in this question is transitive. Transitivity states that if a=b and b=c, then a=c. In this case, it is given that 3 is equal to (9/3) and (9/3) is equal to (30/10). Therefore, by the transitive property, it can be concluded that 3 is equal to (30/10).
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7.
Which of the following is the contrapositive of the given statement: If a figure is a square, then it is a quadrilateral?
A.
If a figure is a quadrilateral, then it is a square.
B.
If a figure is not a quadrilateral, then it is not a square.
C.
If a figure is not a square, then it is not a quadrilateral.
D.
If a figure is a square, then it is not a quadrilateral.
Correct Answer B. If a figure is not a quadrilateral, then it is not a square.
Explanation The contrapositive of a conditional statement switches the hypothesis and conclusion and negates both. In the given statement, the hypothesis is "a figure is a square" and the conclusion is "it is a quadrilateral." The contrapositive of this statement is "If a figure is not a quadrilateral, then it is not a square." This is the correct answer because it follows the pattern of switching and negating the hypothesis and conclusion.
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8.
Which of the following is demonstrating the associative property of multiplication?
A.
2(b+4) = 2b+8
B.
2 x 1= 2
C.
2 + (3+4) = (2+3) + 4
D.
A(b x c) = (a x b)c
Correct Answer D. A(b x c) = (a x b)c
Explanation The given equation, a(b x c) = (a x b)c, demonstrates the associative property of multiplication. This property states that the grouping of numbers being multiplied does not affect the result. In this equation, the numbers a, b, and c are being multiplied in different groupings, but the result is the same on both sides of the equation. This shows that the order of multiplication does not matter, and the equation is an example of the associative property.
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9.
In comparing two fractions, with the same numerator, one said he can subtract numerator from the denominator and the smaller difference is the larger fraction. What kind of reasoning is being defined?
A.
Analogy
B.
Deductive reasoning
C.
Inductive reasoning
D.
Intuition
Correct Answer D. Intuition
Explanation The reasoning being defined in this question is intuition. Intuition is a type of reasoning that relies on instinct, gut feelings, or a sense of knowing without conscious reasoning. In this case, the person is using their intuition to determine that the fraction with the smaller difference between the numerator and denominator is larger. This is not a deductive or inductive reasoning process, as it is not based on logical deduction or generalization from specific instances. It is also not an analogy, as there is no comparison being made between two different situations or concepts.
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10.
Below are Polya’s Steps in Problem Solving. Which of the following follows the correct order of the
steps?
1 - Devise a Plan
2 – Look back
3 - Understand the Problem
4 - Carry out the Plan
A.
1342
B.
3142
C.
3214
D.
4132
Correct Answer B. 3142
Explanation The correct order of Polya's Steps in Problem Solving is as follows: 3 - Understand the Problem, 1 - Devise a Plan, 4 - Carry out the Plan, 2 - Look back. This order ensures that the problem is fully understood before coming up with a plan, and then executing that plan. Finally, reflecting on the solution and checking for any mistakes or improvements is done in the last step.
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11.
Which of the following sets are equivalent?
A.
{1, 2, 3, 4} & {2, 3, 4}
B.
{ 9, 8 , 1} & { -1, 0, 1}
C.
{a, b, c, d} & {1, 2}
D.
All of the above
Correct Answer B. { 9, 8 , 1} & { -1, 0, 1}
Explanation The set {9, 8, 1} & { -1, 0, 1} is equivalent to the set { -1, 0, 1} because they both contain the same elements.
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12.
Given: U = {1, 2, 3, 4, 5, 6 ,7 ,8}; A = {2, 4, 6, 8}; B = {1, 2, 3, 5, 7}; C = { 3, 4, 5, 6}.What is (A U B) U C?
A.
{2, 3, 4, 5, 6, 7}
B.
{1, 2, 3, 4}
C.
{∅}
D.
{1, 2, 3, 4, 5, 6, 7, 8}
Correct Answer D. {1, 2, 3, 4, 5, 6, 7, 8}
Explanation The expression (A U B) U C represents the union of sets A U B and C. Set A U B contains all the elements that are in either set A or set B, so it includes {2, 4, 6, 8, 1, 3, 5, 7}. The union of this set with set C includes all the elements that are in either set A U B or set C, so it includes {2, 4, 6, 8, 1, 3, 5, 7, 3, 4, 5, 6}. However, since there are duplicate elements in this set, they are only counted once, resulting in the set {1, 2, 3, 4, 5, 6, 7, 8}.
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13.
Given: U = {1, 2, 3, 4, 5, 6 ,7 ,8}; A = {2, 4, 6, 8}; B = {1, 2, 3, 5, 7}; C = { 3, 4, 5, 6}.
Find A’ – B?
A.
{2}
B.
{ }
C.
{1, 2}
D.
{1, 3, 5, 7}
Correct Answer B. { }
Explanation The correct answer is an empty set because A' represents the complement of set A, which includes all elements in the universal set U that are not in set A. B represents a set of elements that are in the universal set U but not in set A. Therefore, when we find A' - B, we are looking for elements that are in the complement of set A but not in set B. Since there are no elements in the complement of set A and are not in set B, the answer is an empty set.
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14.
Determine whether the given pair of numbers is a Function or Not. (2, 4), (0, 1), (-1, 3), (3, 5), (2, 6)
A.
Function
B.
Not Function
Correct Answer B. Not Function
Explanation This pair of numbers is not a function because there is more than one output (y-value) for the input (x-value) of 2. In the given pairs, (2, 4) and (2, 6) have the same x-value of 2 but different y-values of 4 and 6 respectively. In a function, each input should have only one output.
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15.
Tell whether each of the following is an Intuition, an Analogy, an Inductive or Deductive Reasoning.
I will sleep early because I have a feeling that I will have a wonderful dream.________
Correct Answer Intuition
Explanation The given statement "I will sleep early because I have a feeling that I will have a wonderful dream" suggests that the decision to sleep early is based on a personal instinct or gut feeling rather than logical reasoning or evidence. This aligns with the concept of intuition, which refers to making decisions or judgments based on instinctive feelings or hunches rather than conscious reasoning. Therefore, the correct answer is Intuition.
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16.
Tell whether each of the following is an Intuition, an Analogy, an Inductive or Deductive Reasoning.
All fruits are nutritious. Apple is a fruit, therefore, it is nutritious.________
Correct Answer Deductive Reasoning
Explanation The given statement follows the pattern of deductive reasoning, which involves drawing a specific conclusion based on general premises or principles. In this case, the general premise is that all fruits are nutritious, and the specific premise is that an apple is a fruit. From these premises, the conclusion is made that the apple is nutritious. Deductive reasoning relies on logical inference and is used to make conclusions that are necessarily true if the premises are true.
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17.
Tell whether each of the following is an Intuition, an Analogy, an Inductive or Deductive Reasoning.
In the given, 1x9 = 9, 2x9 =18, 3 x 9 = 27, 4 x 9 = 36, 5 x 9 = 45. In 18, 1+8 = 9, 2+7 = 9, 2+7 = 9, 3+6 = 9, 4+5 = 9.Therefore, the sum of the digits for the product of a natural number and 9 is 9.________
Correct Answer Inductive Reasoning
Explanation The given explanation suggests that the pattern observed in the multiplication of a natural number with 9 is that the sum of the digits of the product will always be 9. This conclusion is based on the observation of multiple examples and the consistency of the pattern. Inductive reasoning involves making generalizations based on specific observations or patterns, which aligns with the given explanation. Therefore, the correct answer is Inductive Reasoning.
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18.
Tell whether each of the following is an Intuition, an Analogy, an Inductive or Deductive Reasoning.
Ray:Line as Arc: Circle.________
Correct Answer Analogy
Explanation The given statement "Ray:Line as Arc: Circle" is an analogy. An analogy is a type of reasoning that compares two things or ideas that are similar in some way. In this analogy, the relationship between a ray and a line is being compared to the relationship between an arc and a circle. Just as a ray is a part of a line, an arc is a part of a circle. Therefore, the correct answer is analogy.
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19.
Tell whether each of the following is an Intuition, an Analogy, an Inductive or Deductive Reasoning.
Christian drew five different triangles. He got the sum of the measures of the three angles of each triangle. He discovered that the sum of the measures of the three angles of any triangle is 180° .________
Correct Answer Inductive Reasoning
Explanation The given information states that Christian drew five different triangles and found that the sum of the measures of the three angles of any triangle is 180Â°. This is an example of inductive reasoning because it is based on observations of specific cases (the five triangles) and generalizing a pattern or rule (the sum of the angles is always 180Â°) from those observations.
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20.
Tell whether each of the following is an Intuition, an Analogy, an Inductive or Deductive Reasoning.
Independent: explanatory as dependent: response.________
Correct Answer Analogy
Explanation The given statement "Independent: explanatory as dependent: response" is an analogy. An analogy is a type of reasoning where a comparison is made between two things that are similar in certain aspects. In this case, the analogy is being used to compare the relationship between "independent" and "explanatory" with the relationship between "dependent" and "response". Just like "independent" and "explanatory" are related, similarly "dependent" and "response" are related. Therefore, the correct answer is analogy.
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21.
Tell whether each of the following is an Intuition, an Analogy, an Inductive or Deductive Reasoning.
Lhevie’s mother is a very good teacher. Lhevie will also become a very good teacher.________
Correct Answer Intuition
Explanation The given statement is an example of intuition because it is based on a gut feeling or instinct rather than logical reasoning or evidence. The conclusion that Lhevie will also become a very good teacher is not based on any specific evidence or logical reasoning, but rather on the assumption that being raised by a good teacher will automatically make Lhevie a good teacher as well.
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22.
Tell whether each of the following is an Intuition, an Analogy, an Inductive or Deductive Reasoning.
By looking at the color of the ripe mangoes, you can already tell the one sweeter than the other.________
Correct Answer Intuition
Explanation The given statement suggests that the ability to determine the sweetness of mangoes based on their color is a form of intuition. Intuition refers to the ability to understand or know something instinctively, without the need for conscious reasoning or evidence. In this case, the individual is relying on their gut feeling or instinct to judge the sweetness of the mangoes based on their color, rather than using logical reasoning or evidence.
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23.
Tell whether each of the following is an Intuition, an Analogy, an Inductive or Deductive Reasoning.
I know I will win a jackpot on this slot machine in the next 10 tries, because it has not paid out any money during the last 45 tries.________
Correct Answer Intuition
Explanation The given statement is an example of intuition. Intuition is a type of reasoning that relies on personal feelings, instincts, or hunches rather than logical or evidence-based thinking. In this case, the person believes they will win a jackpot on the slot machine based on their gut feeling or intuition, without any concrete evidence or logical reasoning.
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24.
Tell whether each of the following is an Intuition, an Analogy, an Inductive or Deductive Reasoning.
Hat is to head as slippers are to feet.________
Correct Answer Analogy
Explanation The given statement "Hat is to head as slippers are to feet" is an analogy. An analogy is a type of reasoning where a comparison is made between two things that are similar in certain aspects. In this case, the analogy is drawing a similarity between the relationship of a hat to a head and the relationship of slippers to feet. Both pairs have a similar function and are related to each other in terms of usage. Therefore, the correct answer is analogy.
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25.
Determine whether the given pair of numbers is a Function or Not.
(-3, 1), (-2, 2), (-1, 3), (0, 4), (1, 5)
A.
Function
B.
Not Function
Correct Answer A. Function
Explanation The given pair of numbers is a function because each input value (x-coordinate) corresponds to exactly one output value (y-coordinate). In other words, for every x-value, there is only one y-value associated with it.
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26.
The number of elements in power set {1, 2, 3} are
A.
5
B.
6
C.
7
D.
8
Correct Answer D. 8
Explanation The power set of a set is the set of all possible subsets of that set. In this case, the set {1, 2, 3} has 3 elements. To find the number of elements in the power set, we can use the formula 2^n, where n is the number of elements in the original set. Therefore, the power set of {1, 2, 3} will have 2^3 = 8 elements.
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27.
The range of R = {(0,2), (2,4),(3,4),(4, 5)} is
A.
{0, 2, 4, 5}
B.
{0, 2, 3, 4}
C.
{2, 4, 5}
D.
{3, 4 , 5}
Correct Answer C. {2, 4, 5}
Explanation The correct answer is {2, 4, 5} because the range of a set is the set of all possible output values. In this case, the set R has four pairs of values. The first element of each pair represents the input value, and the second element represents the output value. The range is determined by taking all the unique output values from the set R. In this case, the unique output values are 2, 4, and 5. Therefore, the range of R is {2, 4, 5}.
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28.
If 2 sets A and B are given, then the set consisting of all the elements which are either in A or in B or in both is called
A.
Intersection of A and B
B.
Union of A and B
C.
Complement of A
D.
Complement of B
Correct Answer B. Union of A and B
Explanation The set consisting of all the elements which are either in set A or in set B or in both is called the union of A and B.
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29.
Which of the following sets is a universal set for the other four sets?
A.
The set of even natural numbers
B.
The set of odd natural numbers
C.
The set of natural numbers
D.
The set of integers
Correct Answer D. The set of integers
Explanation The set of integers is a universal set for the other four sets because it includes all the elements from each of the other sets. The set of integers includes both even and odd natural numbers, as well as zero, which is not included in the sets of even and odd natural numbers. Additionally, the set of integers includes all the natural numbers, which are a subset of the integers. Therefore, the set of integers is the largest set that contains all the elements from the other four sets.
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30.
Quadrilateral ⊆ polygon
A.
True
B.
False
Correct Answer A. True
Explanation A quadrilateral is a type of polygon, which means it is a closed figure with straight sides. Therefore, the statement that a quadrilateral is a polygon is true.
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31.
Whole numbers ⊆ natural numbers
A.
True
B.
False
Correct Answer B. False
Explanation The statement "Whole numbers ⊆ natural numbers" is false. Whole numbers include all the positive integers, zero, and all the negative integers. Natural numbers, on the other hand, only include the positive integers. Therefore, not all whole numbers are natural numbers, making the statement false.
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32.
{a} ∈ {d, e, f, a}
A.
True
B.
False
Correct Answer B. False
Explanation The statement {a} ∈ {d, e, f, a} is true because the element "a" is indeed present in the set {d, e, f, a}. Therefore, the correct answer is False.
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33.
Natural numbers ⊆ whole numbers
A.
True
B.
False
Correct Answer A. True
Explanation The statement "Natural numbers ⊆ whole numbers" is true. Natural numbers are the set of positive integers starting from 1, while whole numbers are the set of non-negative integers including 0. Since the set of natural numbers is a subset of the set of whole numbers, the statement is correct.
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34.
Integers ⊆ natural numbers
A.
True
B.
False
Correct Answer B. False
Explanation Integers are not a subset of natural numbers. While natural numbers include only positive whole numbers, integers include both positive and negative whole numbers, as well as zero. Therefore, the statement "Integers ⊆ natural numbers" is false.
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35.
0 ∈ ∅
A.
True
B.
False
Correct Answer B. False
Explanation The statement "0 is an element of the empty set" is false. The empty set, also known as the null set, is a set that contains no elements. Since 0 is a number and not an empty set, it cannot be an element of the empty set. Therefore, the answer is false.