GEOMETRIC LOGIC TRANSLATIONS
Translate this statement into symbolic logic: "If a polygon is a square, then it has four equal sides."
Answer: What is "If p, then q" or "p → q" where p = "polygon is a square" and q = "it has four equal sides"?
Is this conjecture true or false: "All prime numbers are odd."
Answer: What is "False" (counterexample: 2 is prime and even)?
In set notation, what does A ∩ B represent?
Answer: What is "the intersection of sets A and B" or "elements that are in both A and B"?
In a Venn diagram with two overlapping circles A and B, what does the overlapping region represent?
Answer: What is "A ∩ B" or "elements that belong to both sets A and B"?
Name the logical rule: "If p then q. p is true. Therefore, q is true."
Answer: What is "Modus Ponens" or "Law of Detachment"?
Write the converse of: "If two lines are parallel, then they never intersect."
Answer: What is "If two lines never intersect, then they are parallel"?
Provide a counterexample for: "If a number ends in 5, then it is divisible by 15."
Answer: What is "25" (or any number ending in 5 that's not divisible by 15, like 35, 65, 85)?
If A = {1, 2, 3} and B = {2, 3, 4}, what is A ∪ B?
Answer: What is "{1, 2, 3, 4}"?
If a Venn diagram shows 15 elements in set A only, 10 elements in both A and B, and 8 elements in set B only, how many total elements are there?
Answer: What is "33 elements"?
What is the contrapositive of "If it rains, then the ground gets wet"?
Answer: What is "If the ground is not wet, then it did not rain"?
Translate: "All right triangles have a 90-degree angle" into conditional form.
Answer: What is "If a triangle is a right triangle, then it has a 90-degree angle"?
True or False: "The sum of two irrational numbers is always irrational."
Answer: What is "False" (counterexample: √2 + (-√2) = 0, which is rational)?
What set operation is represented by the shaded region that includes everything in A but not in B?
Answer: What is "A - B" or "A \ B" (set difference)?
In a three-set Venn diagram (A, B, C), how many distinct regions are possible?
Answer: What is "8 regions"?
Complete this syllogism: "All mammals are warm-blooded. All dogs are mammals. Therefore, ___"
Answer: What is "All dogs are warm-blooded"?
Write the contrapositive of: "If a quadrilateral is a rectangle, then opposite sides are parallel."
Answer: What is "If opposite sides of a quadrilateral are not parallel, then it is not a rectangle"?
Find a counterexample for: "If a quadrilateral has four equal sides, then it is a square."
Answer: What is "a rhombus" (that is not a square)?
If set A is completely inside set B in a Venn diagram, what is the relationship between A and B?
Answer: What is "A is a subset of B" or "A ⊆ B"?
If |A| = 20, |B| = 15, and |A ∩ B| = 5, what is |A ∪ B|?
Answer: What is "30"? (Using the formula |A ∪ B| = |A| + |B| - |A ∩ B|)
Identify the logical fallacy: "If p then q. q is true. Therefore, p is true."
Answer: What is "Affirming the Consequent" (invalid reasoning)?
Express this biconditional in words: "A triangle is equilateral if and only if all three sides are equal."
Answer: What is "A triangle is equilateral if all three sides are equal, AND if a triangle has all three sides equal, then it is equilateral"?
Disprove: "For any triangle, if two sides are equal, then the triangle is equilateral."
Answer: What is "an isosceles triangle that is not equilateral" (counterexample: a triangle with sides 3, 3, 4)?
Given A = {x | x is a multiple of 2} and B = {x | x is a multiple of 4}, describe the relationship between A and B.
Answer: What is "B is a subset of A" or "every element in B is also in A"?
In a survey Venn diagram: 25 people like pizza, 30 like burgers, 15 like both, and 10 like neither. What's the total number surveyed?
Answer: What is "50 people"? (10 pizza only + 15 burger only + 15 both + 10 neither)
Given: "If x > 5, then x > 3" and "x ≤ 3." What can you conclude using contrapositive reasoning?
Answer: What is "x ≤ 5"? (From the contrapositive: "If x ≤ 3, then x ≤ 5")