Predicates and Quantifiers Practice Questions

Predicates and quantifiers are concepts in mathematical logic used to describe properties of objects and make logical statements about them.

• A predicate is a statement that contains one or more variables and becomes true or false when values are assigned to those variables.
• Quantifiers specify how many elements in a set satisfy the predicate, such as "for all" (∀) or "there exists" (∃).

Question 1: Let P(x) be the predicate "x > 5" where x is a real number.

Solution:

P(7) is true because 7 > 5

P(3) is false because 3 is not > 5

Question 2: Let Q(x, y) be the predicate "x + y = 10" where x and y are integers.

Solution:

• Q(3, 7) is true because 3 + 7 = 10
• Q(4, 5) is false because 4 + 5 ≠ 10

Question 3: Let R(x) be the predicate "x² ≥ 0" where x is a real number.

Solution:

The statement ∀x R(x) is true because for all real numbers, their square is always non-negative.

Question 4: Let S(x) be the predicate "x² = 4" where x is a real number.

Solution:

The statement ∃x S(x) is true because there exist real numbers (2 and -2) whose square is 4.

Question 5: Let R(x,y) be the predicate "x < y" where x and y are real numbers.

Solution:

The statement ∀x ∀y R(x,y) is false because it's not true that every real number is less than every other real number.

The statement ∃x ∃y R(x,y) is true because we can find two real numbers where one is less than the other (e.g., 1 < 2).

Question 6: Let Q(x,y) be the predicate "x + y > 10" where x and y are real numbers. Determine the truth value of ∀x ∃y Q(x, y).

Solution:

∀x ∃y Q(x,y) is true.

For any real number x, we can always find a y such that x + y > 10.

We can choose y = 11 - x, which will always make x + y > 10 regardless of the value of x.

Question 7: Let R(x) be the predicate "x is a vowel" where x is a letter in the English alphabet. Write the following statement using predicate logic: "There is a vowel in the English alphabet."

Solution:

The statement can be written as: ∃x R(x)

This reads as "There exists an x such that x is a vowel."

Practice Questions

Question 1: Let Q(x, y) be the predicate "x < y" where x and y are integers. What does ∀x ∃y Q(x,y) mean in words?

Question 2: Let R(x) be the predicate "x is even" where x is an integer. Write the statement "All integers are even" using predicate logic.

Question 3: Let S(x) be the predicate "x is a mammal" and T(x) be "x can fly" where x is an animal. How would you express "Some mammals can fly" using predicate logic?

Question 4: Let P(x, y) be the predicate "x is the parent of y" where x and y are people. Express "Everyone has a parent" using predicate logic.

Question 5: Let Q(x) be the predicate "x is prime" where x is a positive integer. What is the negation of ∀x Q(x)?

Question 6: Let T(x, y) be the predicate "x is taller than y" where x and y are people. Express the statement "Nobody is taller than themselves" using predicate logic.