Let the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, with subsets A = {2, 4, 6, 8, 10} and B = {3, 6, 9}. What is the symmetric difference (A \ B) ∪ (B \ A)?
Matematicas
Question · Easy
Contrapositive, darling: a torch song
Short
Which of the following compound propositions is logically equivalent to the standard conditional statement p → q?
In an applicant pool of 120 students sitting CIDE entrance exams: 65 apply to Economics, 50 to Political Science, and 45 to Law. Exactly 20 apply to both Economics and Political Science, 15 to Political Science and Law, 18 to Economics and Law, and 8 apply to all three programs. How many applicants apply to none of these three programs?
Question · Medium
What is the formal logical negation of the quantified assertion: 'For all x ∈ S, if P(x) is true, then Q(x) is true' (∀x ∈ S, P(x) → Q(x))?
Let A and B be finite sets such that |A| = 4, |B| = 3, and |A ∩ B| = 2. What is the cardinality of the power set of (A ∪ B), denoted |P(A ∪ B)|?
Using De Morgan's laws and principles of propositional equivalence, which of the following expressions is logically equivalent to ¬((p ∧ ¬q) ∨ r)?
Let A and B be arbitrary subsets of a universal set U. Which of the following set identities is universally true for the symmetric difference A Δ B?
Question · Hard
Which of the following propositional formulas is a tautology (evaluates to true under all possible truth value assignments of its variables)?
Consider the following deductive argument in economic analysis: Premise 1: If the central bank increases interest rates, private investment falls. Premise 2: Private investment did not fall. Conclusion: Therefore, the central bank did not increase interest rates. Which valid inference rule does this argument demonstrate?
Let the universal set be the set of all real numbers ℝ. Given the real intervals A = (-∞, 4] and B = (-2, 7), what is the complement in ℝ of their intersection, written (A ∩ B)ᶜ?
Completely factor the algebraic expression: 3x³ - 12x.
Find the complete solution set for the quadratic equation: 2x² - 7x + 3 = 0.
Simplify the rational expression: ((x³ - 8) / (x² - 4)) · ((x + 2) / (x² + 2x + 4)) for all real values of x where the expression is defined.
According to the Polynomial Remainder Theorem, what is the remainder when the polynomial P(x) = 2x⁴ - 3x³ + x - 5 is divided by (x - 2)?
Solve the rational inequality: (2x - 5) / (x + 3) ≤ 1.
Determine all real points of intersection between the line y = 2x + 1 and the parabola y = x² - x - 3.
The cubic polynomial equation x³ - 7x² + 14x - 8 = 0 has three positive real roots r₁, r₂, and r₃ that form a geometric progression with common ratio q > 1. What is the value of the middle root r₂?
Simplify the expression ((16x⁴y⁻²) / (81x⁻⁴y⁶))^(1/4) for all strictly positive real numbers x and y.
Find the complete real solution set to the logarithmic equation: log₂(x + 3) + log₂(x - 1) = 5.
Find the complete solution set in ℝ for the inequality involving absolute value: |3x - 4| > 2x + 1.
Expand completely the binomial expression: (2x - 3)³.
For what nonzero value of k does the quadratic equation kx² - 6x + 2 = 0 have exactly one real root (a real root of multiplicity two)?
Find the complete real solution set to the exponential equation: 3^(2x + 1) - 10 · 3^x + 3 = 0.
For what real value of parameter a does the linear system have infinitely many solutions? x + y + z = 6 2x - y + 3z = 9 x + 4y = a
Suppose A is a subset of B and B ∩ C = ∅. Which statement must be true?
In the Cartesian plane, what are the exact Euclidean distance d and midpoint M between points P(1, -2) and Q(7, 6)?
Find the general equation of the line that passes through the point (2, -1) and is perpendicular to the line 3x - 4y = 8.
A circle is given by the general equation x² + y² - 6x + 4y - 12 = 0. What are the coordinates of its center C(h, k) and the length of its radius r?
Find the standard equation of the parabola with vertex at V(2, 3) and directrix line x = -1.
What is the natural domain in ℝ of the real-valued function f(x) = √((x - 2) / (x + 5))?
Given the bijective rational function f(x) = (3x + 1) / (2x - 5) defined for x ≠ 5/2, find the inverse function f⁻¹(x) and specify its domain.
Given the real functions f(x) = x² + 1 and g(x) = 2x - 3, what is the value of the composite function (f ∘ g)(2)?
Simplify the trigonometric expression: (sin θ) / (1 + cos θ) + (1 + cos θ) / (sin θ) for all angles θ where both terms are defined.
An ellipse centered at the origin is described by 9x² + 16y² = 144. Find its vertices along the major axis and its numerical eccentricity e.
Find the equations of the linear asymptotes for the hyperbola given by: 4x² - 9y² = 36.
Compared to the base graph of y = f(x), how is the graph of y = f(x - 3) + 2 transformed in the Cartesian plane?
Find all exact solutions to the trigonometric equation 2 cos²(x) - cos(x) - 1 = 0 in the interval [0, 2π).
What is the exact perpendicular distance from the point P(3, -2) to the line 5x - 12y + 10 = 0?
Determine the values of constants a and b that make the piecewise function f(x) continuous at x = 1: f(x) = 2x + a for x < 1; f(x) = 5 for x = 1; f(x) = bx² - 3 for x > 1.
Determine whether the real function f(x) = x³ / (x² + 1) is an even function, an odd function, or neither.
What is the equation of the line through the points (-2, 5) and (4, -7)?
Which equation represents a circle with center (-4, 1) and radius 3?
What is the vertex of the parabola y = x² + 4x - 1?
What is the real-number domain of f(x) = √(5 - 2x)?
If f(x) = 3x - 7, which formula gives the inverse function f⁻¹(x)?
At what point do the lines y = 2x + 1 and x + y = 10 intersect?
Which interval is the solution to the inequality |2x - 3| < 5?
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