Texas A&M University (TAMU) MATH 367 Structure of Mathematics III Test 1 Practice

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• Which properties define a partial order?
• There exists a natural bijection between A×B and B×A given by swapping coordinates. What determines the existence of such a bijection?
• In the context of proving P-->Q, which option is not a standard move in a direct proof?
• For a finite group G and subgroup H, the index [G:H] equals
• What is the null (empty) set?
• What is the inverse of the statement 'If it is raining, then it is Monday'?
• For a finite set A, show that |P(A)| = 2^{|A|}. Which statement supports this?
• Which of the following is an example of a ring that is not a field and why?
• What is the truth value of p-->q when p is false and q is true?
• Which expression denotes the implication from p to q?
• In GF(p) with prime p, what can be said about the multiplicative group of nonzero elements?
• In a function f: H -> K, which is the domain?
• Which statement expresses De Morgan's law for negating a conjunction?
• In a direct proof of P-->Q, what is the standard sequence?
• Which of the following is an example of a ring with a multiplicative identity?
• Which statement is a correct De Morgan law for complements of unions and intersections?
• Which statement is a valid characterization of the subset relation ⊆ on P(S)?
• Which statement is true about supplementary angles?
• According to the incidence axioms, which statement about two points and a line is true?
• For finite sets A and B, what is the cardinality of the Cartesian product A×B?
• Suppose A and B are finite sets with A ∩ B = ∅. What is |A ∪ B|?
• Which relation on the integers is a classic example of an equivalence relation?
• If A ⊆ B and B ⊆ A, what can we conclude?
• In any group, the inverse of a product satisfies which equality?
• Negation of the statement ‘Sue works or goes to school every day’ is which of the following?
• What is the complement A^c with respect to a universal set U?
• Which of the following is the correct expression for ker φ?
• If two angles have the same measure, what can be concluded about their congruence?
• If a ≡ b (mod n) and c ≡ d (mod n), which statements hold?
• In proving P implies Q by contrapositive, which implication is proved directly?
• Which of the following numbers is the sum of the digits of 15?
• Given A = {1,2,4,7,9} and B = {2,5,7,8}, which is A ∪ B?
• In a proof by contradiction, what does deriving a contradiction show about the assumed scenario P and not Q?
• What does the First Isomorphism Theorem assert for a homomorphism φ: G → H?
• In the First Isomorphism Theorem, G/ker φ is isomorphic to which algebraic object?
• State the Fundamental Theorem of Algebra for polynomials over the complex numbers.
• Which statement characterizes a bijection?
• Given A = {1,2,4,7,9} and B = {2,5,7,8}, which is A ∩ B?
• What best describes axioms (postulates) in this system?
• Which statement correctly reflects the incidence axiom for two planes?
• The well-ordering principle is equivalent to which statement?
• Which statement demonstrates that Q is dense in R?
• In triangle ABC, given a point P, P is inside angle ABC precisely when P and A lie on the same side of line BC and P and C lie on the same side of line AB. Which option matches this condition?
• Provide a simple bijection between {1,2,3} and {a,b,c}.
• Which description defines triangle ABC?
• When proving by induction that 1+2+...+n = n(n+1)/2, what is the base case?
• In Z_4, what is the order of the element 2?
• Which statement about the real numbers is true?
• If φ: R → S is a ring homomorphism, the kernel of φ is what kind of algebraic object?
• For a finite set A with n elements, which statement about P(A) is true?
• Which properties show that ⊆ on P(S) is a partial order?
• State the Fundamental Theorem of Arithmetic.
• Which statement expresses the reason for the formula |A∪B| = |A|+|B|−|A∩B|?
• Which statement is guaranteed for every finite nonempty partially ordered set (poset)?
• Which of the following statements about the Cantor pairing function is true?
• If two planes are parallel, their intersection is which of the following?
• Let A1, A2, ..., Ak be finite sets. Which statement correctly justifies that their union is finite?
• For the ring homomorphism f: Z → Z/nZ defined by f(a) = a mod n, what is ker f?
• In Cantor's diagonal argument, which property of the constructed set S leads to a contradiction with assuming surjectivity?
• The number of k-element subsets of an n-element set is given by which expression?
• Line segment AB is defined as which of the following?
• Which statement best describes an injective function f: A → B?
• For p = 7, what is the order of the multiplicative group of nonzero elements of GF(p)?
• The composition of two injective functions is:
• From the statement 'It is raining, then it is Monday', select the Contrapositive.
• Which of the following are defined terms in the axiomatic system?
• If A ⊆ B and B ⊆ A, what can be concluded about A and B?
• Which of the following statements expresses the Archimedean property of the real numbers?
• If a universal statement “for all x, P(x) implies Q(x)” is true, which statement is its contrapositive?
• The power set P(X) forms a lattice under which operations relative to subset inclusion?
• What does the Vertical Angles Theorem assert?
• Which of the following are the undefined terms of the axiomatic system?
• Negate ∀x ∈ D P(x). What is the equivalent existential form?
• Which statement correctly defines an injective function?
• Let H ≤ G be finite. Which relation between |G| and |H| holds?
• Let A, B, C be collinear. If AB + BC = AC, which statement must be true?
• A function f: A → B is bijective if and only if
• From the statement 'It is raining, then it is Monday', select the Converse.
• Two angles are supplementary if their measures sum to:
• Which describes the ray AB?
• Left cosets of a subgroup H in a group G have what partitioning property, and what is the meaning of the index [G:H]?
• Which statement best illustrates the rational root test for polynomials with integer coefficients?
• An angle is best described as:
• In the HL postulate, which parts must be congruent to conclude triangle congruence for right triangles?
• If f: A → B and g: B → C are injective, is g ∘ f injective?
• If f: A→B is bijective, which statement about its inverse f^{-1}: B→A is true?
• Congruence modulo 3 on the integers is an equivalence relation because it is
• Which statement is true about the SAS postulate’s conclusion when two triangles satisfy its conditions?
• Verify that the map x ↦ -x on the integers preserves addition; what kind of automorphism is it?
• For a group homomorphism φ: G → H, im φ always contains which element of H?
• In propositional logic, A XOR B is equivalent to which expression using only AND, OR, and NOT?
• Which half-planes define the interior of triangle ABC?
• For finite sets A and B with |A|=m, |B|=n, and |A∩B|=k, what is |A∪B|?
• Which statement about a line is always true?
• Let a_n satisfy a_n = 2 a_{n-1} with a_0 = 1. What is the closed-form expression for a_n?
• Are Q and R countable or uncountable, respectively?
• Segment Construction Theorem: Given segment AB and ray CD, there is exactly one point E on ray CD such that AB ≅ CE. Which statement describes this theorem?
• Which statement is equivalent to H ≤ G (H is a subgroup)?
• Which description matches the equivalence class [a] modulo 3?
• Which statement correctly states that if n^2 is even, then n is even?
• Which statement expresses a consequence of the First Isomorphism Theorem?
• If p is true and q is false, what is the truth value of the disjunction p V q?
• If S is a set with n elements, how many elements does its power set P(S) have?
• Which expression is equal to the complement of the complement of A with respect to a universal set U?
• If |G| = 60 and |H| = 12 with H ≤ G, does such a subgroup exist?
• In the additive group generated by 1, which statement is true about Z?
• If f∘g is injective, what can be concluded about g?
• Let a, b, c be integers with a | b and b | c. Which statement must be true?
• Which condition is part of the subgroup criterion for a subset H of G to be a subgroup?
• In Z_n under addition, the order of [k] is
• Given A = {1,2,4,7,9} and B = {2,5,7,8}, which is A - B?
• According to the incidence axiom discussed, if two distinct planes intersect, their intersection is which geometric object?
• Which classification of a triangle has all three sides congruent?
• In the induction step for the sum 1+2+...+n = n(n+1)/2, if S_n denotes the sum to n, which equation expresses S_{n+1}?
• A triangle with no sides congruent is called what?
• In the additive group Z, which element has finite order?
• What is another standard name for the null set?
• Which number is a valid counterexample to the statement: If the sum of the digits of a number is divisible by 6, then the number is divisible by 6?
• What is the union of ray AB and ray BA?
• If A ⊆ B and B ⊆ A, what can be concluded about A and B?
• Which statement is false?
• Which symbol denotes the negation of a single proposition p?
• Suppose f is an injection from a finite set X to itself. Which statement must be true?
• Which of the following is a field?
• Given a bijection f: A→B, the inverse function f^{-1}: B→A is always:
• A relation on a set is reflexive and symmetric. Is it necessarily transitive?
• According to incidence axiom 2, how many noncollinear points are required to determine a plane?
• Which statement correctly distinguishes a valid argument from a sound argument?
• Which statement about vertical angles is true?
• If A and B are finite sets with |A| = n and |B| = m, how many elements does the Cartesian product A × B have?
• Which statement is always true about A and its complement A^c?
• What is the contrapositive of p → q?
• If AB = CD, what can be concluded about segments AB and CD?
• If A has n elements, how many subsets does P(A) have?
• What does the well-ordering principle for natural numbers assert?
• In the cyclic group Z_n under addition, which element generates the group?
• What can be said about the image of a ring homomorphism φ: R → S?
• A right angle is defined as an angle whose measure is:
• Which statement correctly captures the relationship between injectivity and left inverses?
• Which properties must hold for ≤ on the integers to be a partial order?
• Which statement is the contrapositive of the generic form 'If p then q'?
• Which property is NOT required for a relation to be an equivalence relation on a set?
• Which statement about the null set is true?
• Which statement correctly describes a subset relation?
• In a function notation f: H -> K, which statement about the domain is true?
• According to the First Isomorphism Theorem, if φ: G → H is a group homomorphism, G/ker φ is isomorphic to
• Which statement correctly describes the power set P(S)?
• State Euclid's lemma: If p is prime and p divides ab, what must be true?
• What is the Cantor pairing function π(x,y) that provides a bijection N×N→N?
• In a group G with a subgroup H, what is true about left cosets in relation to [G:H]?
• Which of the following statements is correct about the lattice structure on P(X) with union as join and intersection as meet?
• If p is true and q is false, what is the truth value of the conjunction p^q?
• Which statement best reflects a form of Lagrange's theorem?
• Which statement best defines an equivalence relation?
• Which expression represents the conjunction of two propositions p and q?
• Which method of proving P implies Q requires assuming P is true and showing Q must follow?
• If f∘g is surjective, what can be concluded about f?
• Which expression represents the disjunction of two propositions p and q?
• Incidence axiom 3 says that if P and Q lie in plane E, then what can be said about line PQ?
• Which statement is guaranteed by the ruler postulate?
• Which of the following is not an undefined term in the axiomatic system?
• What is the intersection of ray AB and ray BA?
• Which of the following describes a contrapositive proof for P-->Q?
• What is the negation of the statement 'Some days are not sunny'?
• Using the multiplication principle, if there are m ways to choose the first item and n ways to choose the second item for each first choice, how many total choices are there?
• If two angles form a linear pair, what can we say about their measures?
• Which of the following is the correct De Morgan’s law for the complement of a union and an intersection?
• If 11 objects are placed into 5 boxes, at least one box must contain at least how many objects?
• Negation of the statement ‘Every dog has four legs’ is which of the following?
• Let G be a finite group of order 12. Which of the following statements is always true?
• Which option describes a valid strong induction proof sketch for the formula 1 + 2 + ... + n = n(n+1)/2 for all n ≥ 1?
• Which of the following is NOT a valid triangle congruence postulate?
• ASA postulate states that two angles and which side determine triangle congruence?
• Segment Construction Theorem: For AB and ray CD, which description correctly describes the theorem?
• Which of the following is not an indirect proof method?
• What is the truth value of p-->q when p is true and q is true?
• Surjectivity means what about im(f)?
• Which statement describes the rational root test for polynomials with integer coefficients?
• An acute angle is an angle with measure:
• Incidence axiom 2 states that how many noncollinear points determine a plane?
• In a proof by contradiction, which assumption is made to derive a contradiction?
• What is the sum 1 + 2 + ... + n?
• What is the contrapositive of the statement 'If it is raining, then it is Monday'?
• Which statement about a finite field is true?
• Distinguish between congruence classes modulo n and ordinary remainders.
• In the angle-angle-side postulate, which configuration guarantees congruence of two triangles?
• Which property must hold for an injective function?
• Let φ: G → H be a group homomorphism. Which statement about im φ is true?
• Which statement expresses the supplement postulate?
• What does A ∩ B denote in set notation?
• Using only AND, OR, and NOT, which expression is equivalent to A → B?
• Which of the following is a bijection from a two-element set to a two-element set?
• How many distinct residue classes modulo 3 exist in integers?
• Which is a valid example of a ring Z_n with a nonzero zero divisor when n is composite?
• From the statement 'It is raining, then it is Monday', select the Inverse.
• Which statement best describes the base-case description in a strong induction proof for the sum formula 1 + 2 + ... + n = n(n+1)/2?
• Which statement defines the ASA congruence criterion?
• Which is the power set of {1,2}?
• Which expression is logically equivalent to p implies q?
• Which statement correctly describes the SAS postulate?
• Negation of the statement ‘John is tall and has black hair’ is which of the following?
• Which statement correctly describes an isosceles triangle?
• In Cantor’s diagonal argument, which statement best describes why the diagonal set S is a problem for surjectivity?
• HL postulate applies to which type of triangles?
• If f: A → B is injective, what statement about the image of f is always true?
• Which statement best captures the goal of a direct proof of P-->Q?
• In S3, which pair does not commute?
• If a set S has n elements, how many elements does its power set P(S) have?
• If A, B, C are collinear and AB = BC, what can be concluded about B with respect to A and C?
• Is divisibility relation on positive integers a partial order? Provide justification.
• What is Z_4/{0,2} isomorphic to?
• Which of the following is true about the even integers under addition?
• Incidence axiom 3 restated: If P and Q lie in plane E, then line PQ is a subset of E.
• Which statement about the kernel of a ring homomorphism φ: R → S is true?
• Which subgroup of Z_4 is nontrivial, proper, and normal?
• For a nonzero polynomial f in F[x] of degree n over a field F, how many roots can f have in F?
• If A, B, C are collinear with B between A and C, which equality must hold?
• If f: A → B is a bijection, which statement about its inverse f⁻¹ is true?
• The representatives for residue classes modulo 3 can be taken as which triple?
• Which condition characterizes a subset A of a universal set U?
• Cantor's diagonal argument shows there is no surjection from A onto P(A). Which is the diagonal set D?
• An obtuse angle has a measure that is:
• Which statement describes a triangle with exactly two equal sides?