1. "A skyscraper is a very tall building."
A. Incorrect, the building might be quite tall but not as tall as a skyscraper
B. Correct. When we know a building is a skyscraper, it has to be very tall
C. Incorrect, it might be quite tall but not as tall as a skyscraper
D. Incorrect, all skyscrapers are buildings, so it being a building can't be an optional condition.
2. Statement 1: If a number is positive, then it is greater than zero.
Statement 2: If a number is greater than zero, then it is positive.
A. A number is positive if and only if it is not zero. - Incorrect. As per Statement 1 & Statement 2, the number has be greater than zero to be positive.
B. If a number is positive and greater than zero, then it is not negative. Incorrect. Neither Statement 1 nor Statement 2 talk about negative numbers, hence B option introduces new information and is not correct representation of Statement 1 and Statement 2 combined.
C. A number is positive if and only if it is greater than zero. - Correct. This aptly summarizes Statement 1 & Statement 2
D. If a number is non-negative, then it is positive. - Incorrect. Neither Statement 1 nor Statement 2 talk about non-negative numbers, hence B option introduces new information and is not correct representation of Statement 1 and Statement 2 combined.
Note: While answering such Qs we only consider the information given in the initial statements. While we know that a non-negative number is a positive number, since negative word has not been used in the initial statements, this is new information if presented in the solution
3. "A dog is a puppy if and only if it is young," as a conditional and its converse.
Note: Both conditional and converse statements should definitely include a condition. And the conditional and converse formats of an "P is Q if and only if R" statement are : If P is Q then R AND If P is R then Q
A. Conditional: A dog is a puppy. Converse: A dog is young. - Incorrect as no condition
B. Conditional: A puppy is a dog. Converse: A puppy is young. - Incorrect as no condition
C. Conditional: If a puppy is young, then it is a dog. Converse: If a puppy is a dog, then it is young. - Incorrect, because in the initial statement the conclusion is not being drawn upon whether it is a dog or not.
D. Conditional: If a dog is young, then it is a puppy. Converse: If a dog is a puppy, then it is young. - Correct, as this is in the format indicated in the note.
4. If p represents "x=7" and q represents "x2=49," which symbolic statement is true?
A. p→q - Correct, as p→q means if p is true then q is true. Since we know that p represents x=7 and q represents x²=49, then if x is 7 then definitely square of x is 49. So p→q is correct
B. q→p - Incorrect, as q→p means if q is true then p is true. Since we know that p represents x=7 and q represents x²=49, then if x² is 49 then x can be +7 or -7. Hence, x may not necessarily be +7, so q→p is incorrect
C. p↔q - Incorrect, as p↔q means if p is true then q is true and if q is true then p is true. We know that is x=7, x² = 49, however the other way around is not true. Hence, p↔q is incorrect
D. All statements are true. - Incorrect, as we have seen B and C are false
5. Which biconditional statement is true?
A. A number is divisible by 9 if and only if it ends in 3. - False. We need to test this for if a number is divisible by 9 then it ends with 3 AND if a number ends with 3 then it is divisible by 9. Both the case are not true (18 is divisible by 9 but it doesn't end with 3. 43 ends with 3, but it is not divisible by 9)
B. A quadrilateral is a square if and only if it has four right angles. - False. We need to test this for if a quadrilateral is a square then it has four right angles AND if a quadrilateral has four right angles then it is a square. The first statement is true because all squares have four right angles; however, the second statement is not true as a quadrilateral might have four right angles but may not be a square (can also be a rectangle)
C. The sum of two integers is positive if and only if the two integers are positive. - False. We need to test this for if the sum of two integers is positive then the two integers are positive AND if two integers are positive then their sum is also positive. The first statement is not true since a small negative number when added to a significantly higher positive number leads to a positive number (e.g. -3+7 = 4). The second statement is true as when two positive numbers are added they will always lead to a positive result.
D. A number is an even number if and only if it is divisible by 2. - True. We need to test this for if a number is an even number it is divisible by 2 AND if a number is divisible by 2 then it is an even number. Both the statements are true, as all even numbers are divisible by 2, and there is no number that is divisible by 2 but is not an even number.
