Instantiating Predicates And Evaluating Truth Values Quiz

1) Which of the following is a predicate, not a proposition?

2 + 2 = 4

The Moon orbits Earth

X > 5

Today is Tuesday

A predicate contains variables and does not have a definite truth value until the variables are assigned values. Option C contains the variable x, so it is a predicate. The other options are propositions because they are statements that are either true or false without any variables.

Explanation

A predicate contains variables and does not have a definite truth value until the variables are assigned values. Option C contains the variable x, so it is a predicate. The other options are propositions because they are statements that are either true or false without any variables.

2) “x is even” is a:

Statement with truth value

Predicate without a truth value until x is known

Proposition

Constant function

The statement "x is even" depends on the value of x. Until we know what x is, we cannot assign a truth value to it. Therefore, it is a predicate and not a proposition (which would have a definite truth value). It is not a constant function.

Explanation

The statement "x is even" depends on the value of x. Until we know what x is, we cannot assign a truth value to it. Therefore, it is a predicate and not a proposition (which would have a definite truth value). It is not a constant function.

3) If P(x): “x is prime”, then P(7) is:

Predicate

Proposition

Variable

Function

P(7) is the result of substituting the constant 7 for the variable x in the predicate P(x). Since there are no variables left, P(7) is a statement that is either true or false (true). Therefore, it is a proposition.

Explanation

P(7) is the result of substituting the constant 7 for the variable x in the predicate P(x). Since there are no variables left, P(7) is a statement that is either true or false (true). Therefore, it is a proposition.

4) “x > y” is an example of a:

Unary predicate

Binary predicate

Ternary predicate

Function

A binary predicate is one that takes two arguments. The predicate "greater than" relates two objects, x and y.

Explanation

A binary predicate is one that takes two arguments. The predicate "greater than" relates two objects, x and y.

5) For P(x): “x < 3”, determine the truth of P(5).

True

False

Undefined

Variable-dependent

P(5) means "5

Explanation

P(5) means "5

6) For P(x): “x² ≥ 0”, what is the truth of P(−2)?

True

False

Undefined

Unknown

P(−2) means "(-2)² ≥ 0". Since (-2)² is 4, and 4 ≥ 0, the statement is true.

Explanation

P(−2) means "(-2)² ≥ 0". Since (-2)² is 4, and 4 ≥ 0, the statement is true.

7) Let D be a binary predicate where D(a,b) formalizes “a divides b”. Which matches D(x, y)?

Y divides x

X divides y

Y > x

X + y = z

The predicate D(x,y) is a binary predicate that relates x and y. Option B, "x divides y", is a possible meaning for D(x,y). Option A would be the re ⅴ erse. Option D has three variables, so it doesn't fit a binary predicate.

Explanation

The predicate D(x,y) is a binary predicate that relates x and y. Option B, "x divides y", is a possible meaning for D(x,y). Option A would be the re ⅴ erse. Option D has three variables, so it doesn't fit a binary predicate.

8) Which is the correct instantiation of predicate P(x) for x = 5?

P(5)

P(x) = 5

P(x→5)

P{5}

Instantiation means replacing the variable with a specific value. To instantiate P(x) for x=5, we replace x with 5, resulting in P(5). The other options are incorrect: P(x)=5 is an equation, P(x→5) is not standard notation, and P{5} uses braces instead of parentheses.

Explanation

Instantiation means replacing the variable with a specific value. To instantiate P(x) for x=5, we replace x with 5, resulting in P(5). The other options are incorrect: P(x)=5 is an equation, P(x→5) is not standard notation, and P{5} uses braces instead of parentheses.

9) Which is an open sentence?

5 > 3

Today is Monday

X + 2 = 7

4 is even

An open sentence is one that contains variables and thus does not have a definite truth value. Option C contains the variable x, so it is open. The other options are closed sentences (propositions) because they have no variables and have definite truth values.

Explanation

An open sentence is one that contains variables and thus does not have a definite truth value. Option C contains the variable x, so it is open. The other options are closed sentences (propositions) because they have no variables and have definite truth values.

10) In P(x): “x is a planet”, what is the constant in P(Earth)?

P

X

Earth

Planet

In P(Earth), the variable x has been replaced by the constant "Earth". Therefore, Earth is the constant. P is the predicate symbol, x is the variable (which is replaced), and "planet" is the meaning of the predicate.

Explanation

In P(Earth), the variable x has been replaced by the constant "Earth". Therefore, Earth is the constant. P is the predicate symbol, x is the variable (which is replaced), and "planet" is the meaning of the predicate.

11) P(x): “x is odd.” Which becomes a proposition?

P(x)

X is odd

P(7)

Odd(x)

A proposition is a statement that is either true or false and has no free variables. P(7) replaces the variable x with the constant 7, so it becomes a specific statement: "7 is odd", which is true. The other options either contain variables (A, B, D) or are not standard notation (D).

Explanation

A proposition is a statement that is either true or false and has no free variables. P(7) replaces the variable x with the constant 7, so it becomes a specific statement: "7 is odd", which is true. The other options either contain variables (A, B, D) or are not standard notation (D).

12) For the domain of people, P(x): “x is very tall.” The truth value of P(John) depends on:

Domain definition of ‘ very tall’

Arithmetic

Quantifier used

X’s type

The predicate " very tall" is vague and depends on how we define " very tall" in the context of the domain (people). There is no arithmetic involved, no quantifier in P(John), and the type of x is already Given (people). Therefore, the truth value depends on the definition of ' very tall'.

Explanation

The predicate " very tall" is vague and depends on how we define " very tall" in the context of the domain (people). There is no arithmetic involved, no quantifier in P(John), and the type of x is already Given (people). Therefore, the truth value depends on the definition of ' very tall'.

13) If S(x): “x² = 4”, what makes S(x) true?

X = 1

X = 0

X = 2 or x = −2

X = 4

S(x) is true when x squared equals 4. The solutions to this equation are x=2 and x=-2, because 2²=4 and (-2)²=4. The other options do not satisfy the equation.

Explanation

S(x) is true when x squared equals 4. The solutions to this equation are x=2 and x=-2, because 2²=4 and (-2)²=4. The other options do not satisfy the equation.

14) If R(x): “x is even” and S(x): “x is positive,” what does ¬R(x) ∧ S(x) mean?

X is positive and even

X is positive and odd

X is negative and even

X is negative and odd

¬R(x) means "x is not even", which is equivalent to "x is odd". S(x) means "x is positive". The ∧ symbol means "and". So ¬R(x) ∧ S(x) means "x is odd and x is positive", or "x is positive and odd".

Explanation

¬R(x) means "x is not even", which is equivalent to "x is odd". S(x) means "x is positive". The ∧ symbol means "and". So ¬R(x) ∧ S(x) means "x is odd and x is positive", or "x is positive and odd".

15) The predicate Q(x,y) represents what type of relation?

Nullary

Unary

Binary

Ternary

The predicate Q(x,y) takes two arguments, x and y. A predicate with two arguments is called a binary predicate. Nullary predicates take no arguments, unary take one, and ternary take three.

Explanation

The predicate Q(x,y) takes two arguments, x and y. A predicate with two arguments is called a binary predicate. Nullary predicates take no arguments, unary take one, and ternary take three.