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Most birds can fly. Penguins are birds. True or False, can penguins fly? | ¬∀X (bird(X) → flies(X))
bird(Penguin)
¬flies(Penguin) | False |
Mammals produce milk. Bats are mammals. True or False, do bats produce milk? | ∀X (mammal(X) → produces_milk(X))
mammal(Bat) | True |
Fish live in water. Tuna is a fish. True or False, does tuna live in water? | ∀X (fish(X) → lives_in_water(X))
fish(Tuna) | True |
Some animals have four legs. Snakes are animals. True or False, do snakes have four legs? | ¬∀X (animal(X) → has_four_legs(X))
animal(Snake)
¬has_four_legs(Snake) | False |
Triangles have three sides. A given shape is a triangle. True or False, does it have three sides? | ∀X (triangle(X) → three_sides(X))
triangle(Shape) | True |
Some metals are magnetic. Aluminum is a metal. True or False, is aluminum magnetic? | ¬∀X (metal(X) → magnetic(X))
metal(Aluminum)
¬magnetic(Aluminum) | False |
Right angles measure 90 degrees. An angle measures 90 degrees. True or False, is it a right angle? | ∀X (right_angle(X) → measures_90(X))
measures_90(Angle) | True |
All squares are rectangles. A given shape is a square. True or False, is it also a rectangle? | ∀X (square(X) → rectangle(X))
square(Shape) | True |
Some reptiles lay eggs. A crocodile is a reptile. True or False, does it lay eggs? | ∃X (reptile(X) ∧ lays_eggs(X))
reptile(Crocodile) | True |
Quadrilaterals have four sides. A given shape is a quadrilateral. True or False, does it have four sides? | ∀X (quadrilateral(X) → four_sides(X))
quadrilateral(Shape) | True |
All mammals breathe oxygen. A dolphin is a mammal. True or False, does a dolphin breathe oxygen? | ∀X (mammal(X) → breathes_oxygen(X))
mammal(Dolphin) | True |
Some plants require sunlight. Mushrooms are not plants. True or False, do mushrooms require sunlight? | ¬∀X (plant(X) → requires_sunlight(X))
¬plant(Mushroom) | False |
All planets orbit a star. Earth is a planet. True or False, does Earth orbit a star? | ∀X (planet(X) → orbits_star(X))
planet(Earth) | True |
Some numbers are prime. 4 is a number. True or False, s 4 prime? | ¬∀X (number(X) → prime(X))
number(4)
¬prime(4) | False |
The sum of two even numbers is always even. 6 and 8 are even numbers. True or False, is their sum even? | ∀X ∀Y (even(X) ∧ even(Y) → even(sum(X,Y)))
even(6)
even(8) | True |
Electricity conducts through some metals. Gold is a metal. True or False, does gold conduct electricity? | ∃X (metal(X) ∧ conducts_electricity(X))
metal(Gold) | True |
Every square has equal sides. A given shape is a square.True or False, does it have equal sides? | ∀X (square(X) → equal_sides(X))
square(Shape) | True |
If a shape is a polygon, it has at least three sides. A pentagon is a polygon. True or False, does it have at least three sides? | ∀X (polygon(X) → at_least_three_sides(X))
polygon(Pentagon) | True |
Every rectangle has four sides. A square is a rectangle. True or False, does a square have four sides? | ∀X (rectangle(X) → four_sides(X))
rectangle(Square) | True |
All living things require water to survive. A tree is a living thing. True or False, does a tree require water to survive? | ∀X (living_thing(X) → requires_water(X))
living_thing(Tree) | True |
Not all metals are magnetic. Copper is a metal. True or False, is copper magnetic? | ¬∀X (metal(X) → magnetic(X))
metal(Copper)
¬magnetic(Copper) | False |
The sum of angles in a triangle is always 180 degrees. A given shape is a triangle. True or False, does the sum of its angles equal 180 degrees? | ∀X (triangle(X) → sum_of_angles(X, 180))
triangle(Shape) | True |
If x is an even number and even numbers are divisible by 2, is x divisible by 2? | ∀x (even(x) → x mod 2 = 0) | x mod 2 = 0 |
If x > y and y > z, is x > z? | ∀x, y, z ((x > y ∧ y > z) → x > z) | x > z |
If p → q and q → r, does p → r? | ∀p, q, r ((p → q) ∧ (q → r) → (p → r)) | p → r |
If a function f(x) is continuous and differentiable, is it also continuous? | ∀f (differentiable(f) → continuous(f)) | continuous(f) |
If 2x = 10, what is x? | 2x = 10 → x = 10 / 2 | x = 5 |
If a set A has n elements, how many subsets does it have? | |P(A)| = 2^n | 2^n |
If a^m * a^n = a^p, what is p? | a^m * a^n = a^(m+n) | p = m + n |
If x is a real number and x^2 = 9, what are the possible values of x? | x = ±√9 | x = ±3 |
If f(x) = 3x + 2 and g(x) = x - 5, what is f(g(x))? | f(g(x)) = 3(x - 5) + 2 | f(g(x)) = 3x - 13 |
If log(a) + log(b) = log(c), what is c? | log(a) + log(b) = log(ab) | c = ab |
If A ⊆ B and B ⊆ C, does A ⊆ C? | ∀A, B, C (A ⊆ B ∧ B ⊆ C → A ⊆ C) | A ⊆ C |
If f(x) = x^2 and f'(x) is the derivative of f(x), what is f'(x)? | f'(x) = d/dx (x^2) | f'(x) = 2x |
If the determinant of a matrix A is 0, is A invertible? | det(A) = 0 → ¬invertible(A) | ¬invertible(A) |
If x^3 = 27, what is x? | x = ³√27 | x = 3 |
If a system of linear equations has a unique solution, what is the determinant of its coefficient matrix? det(A) ≠ ? | unique_solution → det(A) ≠ 0 | det(A) ≠ 0 |
If x is a prime number greater than 2, True or False, is x odd? | ∀x (prime(x) ∧ x > 2 → odd(x)) | True |
If x ≡ y (mod n), is (x + a) ≡ (y + a) (mod n)? | ∀x, y, a, n (x ≡ y (mod n) → (x + a) ≡ (y + a) (mod n)) | (x + a) ≡ (y + a) (mod n) |
If A and B are independent events, is P(A ∩ B) = P(A)P(B)? | independent(A, B) → P(A ∩ B) = P(A)P(B) | P(A ∩ B) = P(A)P(B) |
If x^4 = 81, what are the possible values of x? | x = ±⁴√81 | x = ±3 |
If a sequence is arithmetic and has first term a and common difference d, what is the nth term? | T_n = a + (n-1)d | T_n = a + (n-1)d |
If sin²θ + cos²θ = 1, what is sin²θ? | sin²θ = 1 - cos²θ | sin²θ = 1 - cos²θ |
If x^5 = 32, what is x? | x = ⁵√32 | x = 2 |
If det(A) ≠ 0, what is A⁻¹? | A⁻¹ = adj(A) / det(A) | A⁻¹ = adj(A) / det(A) |
If a sequence is geometric with first term a and common ratio r, what is the nth term? | T_n = a * r^(n-1) | T_n = a * r^(n-1) |
If the sum of an infinite geometric series is S = a / (1 - r), what is S for a = 3 and r = 0.5? | S = 3 / (1 - 0.5) | S = 6 |
If the derivative of f(x) = e^x is f'(x), what is f'(x)? | d/dx (e^x) = e^x | f'(x) = e^x |
Subsets and Splits