1.
One of the De Morgan’s laws says that the negation of AND statement is logically equivalent to the __________ statement in which each component is negated.
Correct Answer
A. OR
Explanation
De Morgan's law states that the negation of an AND statement is logically equivalent to the OR statement in which each component is negated. This means that if we have a statement like "A AND B", the negation of this statement would be "NOT A OR NOT B". Therefore, the correct answer is OR.
2.
A contradiction is a statement that is always ____. a)true b)False c)None
Correct Answer
B. False
Explanation
A contradiction is a statement that is always false.
3.
(A U B) ∩ C = (A ∩ C) U (B ∩ C) The given statement is
Correct Answer
A. True
Explanation
The given statement is true because the intersection of the union of two sets A and B with set C is equal to the union of the intersection of set A with C and the intersection of set B with C. This can be proven using set theory properties, specifically the distributive property of intersection over union.
4.
Which of the following law states ((A implies B) and (B implies C)) implies (A implies C)
Correct Answer
A. Transitive Law
Explanation
The correct answer is Transitive Law. This law states that if A implies B and B implies C, then A implies C. In other words, if two statements are true and one implies the other, then the first statement also implies the third statement. This law is commonly used in logic and mathematics to establish logical relationships between statements.
5.
If x is even the square of x when added to 3 becomes odd. This statement is
Correct Answer
A. True
Explanation
If x is even, it means that x can be represented as 2n, where n is an integer. If we square x, we get (2n)^2 = 4n^2. When we add 3 to this, we get 4n^2 + 3. Since n^2 is an integer, 4n^2 is always even. Adding an odd number (3) to an even number always results in an odd number. Therefore, if x is even, the square of x when added to 3 becomes odd, making the statement true.
6.
An ordered pair (x,y) in Cartesian product A X B where x is related to y such that xRy, then
Correct Answer
A. Set A is called domain of R and set B is called co-domain of R
Explanation
The correct answer is "Set A is called domain of R and set B is called co-domain of R." This is because in a Cartesian product, the first element of each ordered pair is considered the domain and the second element is considered the co-domain. In this case, x is related to y, so x is the domain (set A) and y is the co-domain (set B).
7.
A X B = B X A where a belongs to A and b belongs to B is True
Correct Answer
A. When a=b
Explanation
The given statement "A X B = B X A where a belongs to A and b belongs to B" is true when a=b. This means that for the two sets A and B, if an element from set A is equal to an element from set B, then the product of A and B will be equal to the product of B and A.
8.
Which of the following is True?
Correct Answer
C. Both
Explanation
Every set is a subset of itself because every element in a set is also an element of itself. Similarly, the null set is a subset of every set because the null set does not contain any elements, and therefore, all the elements of the null set are also elements of any other set.
9.
Collection of all the possible subsets of a given set is called
Correct Answer
A. Power set
Explanation
The collection of all possible subsets of a given set is called the power set. It includes the empty set and the set itself as subsets, as well as all possible combinations of elements from the original set. The power set has a total of 2^n subsets, where n is the number of elements in the original set.
10.
A barber shaves those men who do not shave themselves. Does the barber shave themselves? This is an example of
Correct Answer
A. Russell’s Paradox
Explanation
Russell's Paradox is a logical paradox that arises when considering the set of all sets that do not contain themselves as a member. In this scenario, the statement "the barber shaves those men who do not shave themselves" leads to a contradiction. If the barber shaves themselves, they would be a man who shaves themselves and therefore should not be shaved by the barber. On the other hand, if the barber does not shave themselves, they would be a man who does not shave themselves and therefore should be shaved by the barber. This contradiction highlights the paradoxical nature of the statement and aligns with Russell's Paradox.
11.
A conditional statement and its inverse are
Correct Answer
A. Not logically equivalent.
Explanation
A conditional statement and its inverse are not logically equivalent because they have opposite truth values. In a conditional statement, if the antecedent is true, then the consequent must also be true. However, in its inverse, the antecedent and consequent are negated, resulting in a different truth value. Therefore, the two statements are not equivalent.
12.
If p and q are statements variables, then the bi-conditional of p and q is
Correct Answer
A. P if and only if q
Explanation
The bi-conditional of p and q is represented by "p if and only if q". This means that p is true if and only if q is true. In other words, p and q have the same truth value. If p is true, then q is true, and if q is true, then p is true.
13.
If p and q are statements then which is True?
Correct Answer
C. Both
Explanation
The correct answer is Both. "p is a sufficient condition for q" means that if p is true, then q must also be true. "p is a necessary condition for q" means that if p is not true, then q cannot be true either. Both statements are true because if p is a sufficient condition for q, then p is also a necessary condition for q.
14.
If all the premises are true then the conclusion must be true. Such argument is said to be
Correct Answer
A. Valid.
Explanation
If all the premises in an argument are true, then the conclusion must also be true. This is the definition of a valid argument. In a valid argument, the conclusion logically follows from the premises, meaning that if the premises are true, the conclusion cannot be false. Therefore, the given answer "Valid" is correct.
15.
All actors are robots. Tom Cruise is a robot. Therefore, Tom Cruise is an actor. This is an example of
Correct Answer
A. Invalid argument
Explanation
This is an example of an invalid argument because even though all actors are robots, it does not necessarily mean that all robots are actors. Therefore, the conclusion that Tom Cruise is an actor cannot be logically inferred from the given premises.
16.
All humans are polite can be written as
Correct Answer
C. Both
Explanation
The correct answer is "both" because the statement "All humans are polite" can be expressed using the universal quantifier "For all human beings x" and the predicate "x is polite." Additionally, the set H is mentioned as the set of human beings in the given explanation.
17.
There exists x which belongs to D such that Q(x) is defined to be true
Correct Answer
A. if and only if Q(x) is true for at least one x in D
Explanation
The correct answer is "if and only if Q(x) is true for at least one x in D." This means that there is at least one value of x in the domain D for which the statement Q(x) is true. It does not require all values of x in D to satisfy Q(x), only at least one.
18.
For all rectangles t, t has four sides. Convert Formal to Informal
Correct Answer
A. All rectangles have four sides.
Explanation
The given answer "All rectangles have four sides" is correct because it accurately represents the statement "For all rectangles t, t has four sides." This statement implies that every rectangle, without exception, has four sides.
19.
In a statement containing both for all and there exists , changing the order of quantifier usually
Correct Answer
A. Changes the meaning of the statement.
Explanation
In a statement containing both "for all" and "there exists", changing the order of the quantifiers usually changes the meaning of the statement. This is because the order of quantifiers determines the scope of the variables and the relationship between them. If we change the order, we may end up with a different statement that asserts something different about the variables involved. Therefore, changing the order of quantifiers can alter the meaning of the statement.
20.
Determine which of the following statement is true. Given D={-48, -14, -8, 0, 1, 3, 16, 23, 26, 32, 36}
Correct Answer
A. There exists x that belongs to D , if x is odd then x>0
Explanation
The correct answer is "There exists x that belongs to D , if x is odd then x>0". This statement is true because if we look at the set D, we can see that there are several odd numbers present such as -14, -8, 1, 3, 23, 32, and 36. Among these odd numbers, all of them are greater than 0. Therefore, the statement holds true.
21.
A statement (or proposition) is a sentence that is
Correct Answer
A. True or false but not both.
Explanation
A statement (or proposition) is a sentence that can be evaluated as either true or false, but not both simultaneously. This means that a statement cannot be both true and false at the same time. It must be either true or false, but not a combination of both. This is a fundamental principle in logic and is used to determine the validity and truth value of arguments and statements.
22.
Write the negation –“some girls are sincere.”
Correct Answer
A. Every girl is not sincere.
Explanation
The correct answer is "Every girl is not sincere." This statement is the negation of the given statement "Some girls are sincere." The original statement implies that there is at least one girl who is sincere, while the negation states that it is not the case that every girl is sincere. This means that there could be no sincere girls or only a few sincere girls, contradicting the original statement.
23.
Write the negation –“All men are animals”.
Correct Answer
A. There exists a man who is not an animal.
Explanation
The correct answer is "there exists a man who is not an animal." The original statement "All men are animals" is negated by stating that there exists at least one man who is not an animal. This means that it is not true to say that all men are animals because there is at least one counterexample of a man who is not an animal.
24.
Write the negation –“ I will have tea or coffee.”
Correct Answer
A. I will not have tea and coffee.
Explanation
The correct answer is "I will not have tea and coffee." This is the negation of the statement "I will have tea or coffee." The original statement states that the person will have either tea or coffee, while the correct answer states that the person will not have both tea and coffee.
25.
“n is divisible by d” can be written as
Correct Answer
C. Both
Explanation
The statement "n is divisible by d" can be written as both "n is a multiple of d" and "d is a factor of n". This means that n can be evenly divided by d, resulting in a whole number. In other words, n can be expressed as the product of d and another integer, and d can evenly divide into n without leaving a remainder.
26.
For all integers and b, if a and b are positive and ‘a’ divides ‘b’, then
Correct Answer
A. A less than or equal to b
Explanation
If "a" divides "b", it means that "b" is divisible by "a" without leaving a remainder. In other words, "b" can be expressed as a multiple of "a". Since "a" is a positive integer and "b" is a multiple of "a", it follows that "b" must be greater than or equal to "a". Therefore, the correct answer is "a less than or equal to b".
27.
For all integers a and b then, if a/b and b/a then
Correct Answer
A. A=b
Explanation
If a/b and b/a, it means that the fraction a/b is equal to the fraction b/a. This can only happen if both fractions are equal to 1. In other words, a/b = 1 and b/a = 1. The only way for this to be true is if a = b. Therefore, the correct answer is a=b.
28.
Given any integer n and positive integer d, there exist unique integer q and r such that n=dq + r and
Correct Answer
A. 0 less than or equal to r less than or equal to d
Explanation
The statement "0 less than or equal to r less than or equal to d" is the correct answer because it represents the condition for the remainder (r) when n is divided by d. The remainder can range from 0 to d, inclusive, which means it can be any non-negative integer less than or equal to d. This statement accurately describes the relationship between n, d, q, and r in the given equation n = dq + r.
29.
The floor value of 9.14 is?
Correct Answer
A. 9
Explanation
The floor value of a number refers to the largest integer that is less than or equal to that number. In this case, the floor value of 9.14 is 9 because it is the largest integer that is less than or equal to 9.14.
30.
The sum of any rational number and any irrational number is?
Correct Answer
A. Rational
Explanation
The sum of any rational number and any irrational number is always a rational number. This is because when we add a rational number to an irrational number, the result can always be expressed as a fraction, where the numerator and denominator are both integers. Therefore, the sum will always be a rational number.
31.
The gcd(18,12)=
Correct Answer
A. 6
Explanation
The greatest common divisor (gcd) of two numbers is the largest number that divides both of them without leaving a remainder. In this case, the gcd of 18 and 12 is 6 because 6 is the largest number that can evenly divide both 18 and 12.
32.
For all integers m and n, m+n and m-n are:
Correct Answer
A. Either both odd or both even.
Explanation
For any two integers, if their sum is even, it means that they have the same parity (either both even or both odd). This is because adding an even number to an even number gives an even number, and adding an odd number to an odd number gives an even number. Similarly, if their difference is even, it also means that they have the same parity. Therefore, for all integers m and n, m+n and m-n will either both be odd or both be even.
33.
The ceiling value of 25/4 is?
Correct Answer
A. 7
Explanation
The ceiling value of a number is the smallest integer that is greater than or equal to the number. In this case, 25 divided by 4 is equal to 6.25. Since the ceiling value needs to be greater than or equal to 6.25, the correct answer is 7.
34.
The floor of -2.01 is?
Correct Answer
A. -3
Explanation
The floor of a number is the largest integer that is less than or equal to the given number. In this case, the given number is -2.01. The largest integer that is less than or equal to -2.01 is -3. Therefore, the floor of -2.01 is -3.
35.
The ceiling of 0.999 is?
Correct Answer
A. 1
Explanation
The ceiling function rounds a number up to the nearest integer. In this case, the number 0.999 is already less than 1, so rounding it up will result in the number 1.
36.
For all real numbers x and all integers m, floor value of ( x + m) =
Correct Answer
A. floor of x + m.
Explanation
The floor value of a number x is the largest integer that is less than or equal to x. In this question, we are asked to find the floor value of (x + m), where x is a real number and m is an integer. The correct answer is "floor of x + m" because the floor value of a sum of two numbers is equal to the sum of their floor values. Therefore, we can find the floor of x, add it to m, and take the floor value of their sum to get the overall floor value of (x + m).
37.
The square root of 2 is
Correct Answer
A. Irrational
Explanation
The square root of 2 is irrational because it cannot be expressed as a fraction or a terminating or repeating decimal. It is a non-repeating, non-terminating decimal that goes on forever without a pattern. This can be proven through mathematical methods such as the proof by contradiction or by using the decimal representation of the square root of 2.
38.
If Zeke is a cheater, then Zeke sits in the back row.Zeke sits in the back row.∴ Zeke is a cheater. Is this statement is
Correct Answer
A. Invalid
Explanation
The given statement is invalid because it commits the fallacy of affirming the consequent. Just because Zeke sits in the back row does not necessarily mean that he is a cheater. There could be other reasons why he chooses to sit in the back row, such as preferring a better view or wanting to avoid distractions. Therefore, the statement does not provide enough evidence to conclude that Zeke is a cheater.
39.
The negation of “if p then q” is logically equivalent to?
Correct Answer
A. “p and not q.”
Explanation
The negation of "if p then q" is logically equivalent to "p and not q" because if the statement "if p then q" is false, it means that p is true and q is false. Therefore, the negation of this statement would be "p and not q" which represents the condition where p is true and q is false.
40.
The factorial of zero is?
Correct Answer
B. 1
Explanation
The factorial of zero, denoted as "0!" (read as "zero factorial"), is defined as equal to 1.
41.
According to De Morgan’s laws, the negation of
p: 'Jim is tall and Jim is thin ' is
Correct Answer
A. ∼p: Jim is not tall or Jim is not thin
Explanation
According to De Morgan's laws, the negation of a conjunction is the disjunction of the negations of the individual statements. In this case, the original statement "Jim is tall and Jim is thin" is represented as p. The negation of p is represented as ∼p. Therefore, the correct answer is ∼p: Jim is not tall or Jim is not thin, as it follows the De Morgan's laws.