MA8351-DM UNIT - 1

Let P (x) denote the statement “x >7.” Which of these have truth value true?

• P (0)
• P (4)
• P (6)
• P (9)

The statement, “At least one of your friends is perfect”. Let P (x) be “x is perfect” and let F (x) be “x is your friend” and let the domain be all people.

• ∀x (F (x) → P (x))
• ∀x (F (x) ∧ P (x))
• ∃x (F (x) ∧ P (x))
• ∃x (F (x) → P (x))

Let domain of m includes all students, P (m) be the statement “m spends more than 2 hours in playing polo”. Express ∀m ¬P (m) quantification in English.

• A student is there who spends more than 2 hours in playing polo
• There is a student who does not spend more than 2 hours in playing polo
• All students spends more than 2 hours in playing polo
• No student spends more than 2 hours in playing polo

Translate ∀x∃y(x < y) in English, considering domain as a real number for both the variable.

• For all real number x there exists a real number y such that x is less than y
• For every real number y there exists a real number x such that x is less than y
• For some real number x there exists a real number y such that x is less than y
• For each and every real number x and y such that x is less than y

“The product of two negative real numbers is not negative.” Is given by?

• ∃x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))
• ∃x ∃y ((x < 0) ∧ (y < 0) ∧ (xy > 0))
• ∀x ∃y ((x < 0) ∧ (y < 0) ∧ (xy > 0))
• ∀x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))

Let L(x, y) be the statement “x loves y,” where the domain for both x and y consists of all people in the world. Use quantifiers to express, “Joy is loved by everyone.”

• ∀x L(x, Joy)
• ∀y L(Joy,y)
• ∃y∀x L(x, y)
• ∃x ¬L(Joy, x)

Let T (x, y) mean that student x likes dish y, where the domain for x consists of all students at your school and the domain for y consists of all dishes. Express ¬T (Amit, South Indian) by a simple English sentence.

• All students does not like South Indian dishes.
• Amit does not like South Indian people.
• Amit does not like South Indian dishes.
• Amit does not like some dishes.

Express, “The difference of a real number and itself is zero” using required operators.

• ∀x(x − x! = 0)
• ∀x(x − x = 0)
• ∀x∀y(x − y = 0)
• ∃x(x − x = 0)

Find a counterexample of ∀x∀y(xy > y), where the domain for all variables consists of all integers.

• x = -1, y = 17
• x = -2 y = 8
• Both x = -1, y = 17 and x = -2 y = 8
• Does not have any counter example

Which rule of inference is used in each of these arguments, “If it is Wednesday, then the Smartmart will be crowded. It is Wednesday. Thus, the Smartmart is crowded.”

• Modus tollens
• Modus ponens
• Disjunctive syllogism
• Simplification