Reviewer

What does this mean: ∃ x ∈ ℝ ∧ x ∉ ℚ?

• Some real numbers are not rational

• All real numbers are rational

• Real numbers and rational numbers are the same

• No real number can be irrational

Answer explanation

There exist irrational numbers like √2 and π, which are real but not rational.

Which of the following is a valid conclusion from this: x ∈ ℤ ∧ x ∉ ℕ?

• x is positive

• x is a real number

• x is a natural number

• x must be a fraction

Answer explanation

If x is an integer (ℤ), then it is automatically a real number (ℝ), even if it’s not a natural number (ℕ).

Which statement best represents this logic: If a number is natural, it must also be real?

• ∃ x ∈ ℕ, x ∉ ℝ

• ∀ x ∈ ℝ, x ∈ ℕ

• ∀ x ∈ ℕ, x ∈ ℝ

• ℕ ⊂ ℚ

Answer explanation

All natural numbers (1, 2, 3, …) are real numbers, so the statement is true.

Which expression is logically equivalent to: “If x is rational, then x is real”?

• x ∈ ℝ ⇒ x ∈ ℚ

• x ∈ ℚ ⇒ x ∈ ℝ

• x ∈ ℝ ∧ x ∈ ℚ

• x ∉ ℝ ⇒ x ∉ ℚ

Answer explanation

This reads, “If x is in ℚ (rational), then x is in ℝ (real).” That’s correct because all rational numbers are real numbers.

Which sentence means "All integers are also real numbers"?

• ∃ x ∈ ℤ ⇒ x ∈ ℝ

• ∀ x ∈ ℤ, x ∈ ℝ

• x ∈ ℝ ⇒ x ∈ ℤ

• ∀ x ∈ ℝ, x ∈ ℤ

Answer explanation

All integers are also real numbers, so this statement is correct.

What does the statement x ∈ ℚ ∧ x ∉ ℕ mean?

• x is a rational number and a natural number

• x is not a number

• x is a rational number but not a natural number

• x is a natural number but not rational

Answer explanation

This means x could be a fraction like ½ or a negative number—still rational, but not natural.

What does this mean?

∃ x ∈ ℤ ∧ x < 0

• All real numbers are natural

• All natural numbers are real

• Only some natural numbers are real

• Natural numbers are not real

Answer explanation

This says “for all x in the set of natural numbers, x is also in the set of real numbers.”

Statement: ∀ x ∈ ℤ, x² ≥ 0
This means:

• Some squares of integers are negative

• All integers are positive

• The square of any integer is never negative

• All integers are zero

Answer explanation

Any integer squared results in zero or a positive number.

What is the next number in the sequence?

• 44

• 39

• 30

• 29

Answer explanation

Each number is 2×previous number − 1:
3 = 2×2−1, 5 = 2×3−1, 9 = 2×5−1, etc.

What comes next?
1, 4, 18, 96, ___

• 540

• 600

• 576

• 624

Answer explanation

Multiply by increasing integers:
1×4 = 4, 4×4.5 = 18, 18×5.33... = 96, 96×6 = 576.