Higher Mathematics

1. Build a true statement about a logic term.

Answer: A statement that is always true is a tautology

A proposition that is true under every assignment of its variables is a tautology.

2. The statement P OR Q is false only when both P and Q are false.

Answer: True

True. An OR is true if at least one part is true, so it is false exactly when both parts are false.

3. By De Morgan's law, NOT(P AND Q) is the same as NOT P ____ NOT Q.

• ORcorrect
• AND
• implies
• NOT

De Morgan's law flips AND to OR under a negation, so NOT(P AND Q) equals NOT P OR NOT Q.

4. How many ways can you choose 2 items from 6, that is C(6,2)?

Answer: 15

C(6,2) = 6! / (2! * 4!) = 720 / (2 * 24) = 15.

5. Match each logical connective to its everyday word.

Answer: Conjunction = AND; Disjunction = OR; Negation = NOT

Conjunction is AND, disjunction is OR, and negation is NOT.

6. Which of these are logical connectives? (Select all that apply.)

• Conjunction (AND)correct
• Disjunction (OR)correct
• Negation (NOT)correct
• Factorial

Conjunction (AND), disjunction (OR), and negation (NOT) are connectives; factorial is an arithmetic operation.

7. Build a true statement about disproving a general claim.

Answer: A single counterexample disproves a universal statement

One example that breaks a for-all claim is a counterexample, and it disproves the whole statement.

8. An axiom must be proven before it can be used.

Answer: False

False. An axiom is a starting assumption accepted as true without proof; theorems are what get proved.

9. In a proof by contradiction, you begin by assuming the ____ of what you want to prove.

• oppositecorrect
• same
• converse
• square

You assume the opposite (the negation), then show it forces an impossibility.

10. In a proof by contradiction, what do you assume at the start?

• That the statement is false, the opposite of the goal
• That the statement is already true
• A single random example
• Nothing at all

You assume the statement is false (the opposite of your goal) and then show that this leads to an impossibility.

11. Which of these does NOT prove that a statement is true for ALL cases?

• Checking a single examplecorrect
• Direct proof
• Proof by induction
• Proof by contradiction

Checking one example cannot cover every case; direct proof, induction, and contradiction can each prove a universal claim.

12. Which of these are valid ways to prove a statement? (Select all that apply.)

• Direct proofcorrect
• Proof by inductioncorrect
• Proof by contradictioncorrect
• Assuming the conclusion is true

Direct proof, induction, and contradiction are valid; assuming the conclusion you want is circular reasoning, a fallacy.

13. Build a true statement about the imaginary unit i.

Answer: The square of i equals negative one

By definition i^2 = -1, so the square of i equals negative one.

14. Which of these number systems are fields? (Select all that apply.)

• The rationals Qcorrect
• The reals Rcorrect
• The integers Z
• The naturals N

The rationals Q and the reals R are fields; the integers Z and naturals N are not, since they lack multiplicative inverses.

15. The identity element for addition is 0.

Answer: True

True. Adding 0 to any number leaves it unchanged, which is exactly what an additive identity does.

16. The set of all ____ (like -2, 0, and 5) is written Z.

• integerscorrect
• primes
• fractions
• decimals

Z stands for the integers, the whole numbers together with their negatives and zero.

17. In the complex numbers, what is i^2 (i being the imaginary unit)?

• -1correct
• 1
• i
• -i

The imaginary unit is defined so that i^2 = -1, which is what makes complex numbers work.

18. Three of these number systems are fields. Which one is NOT a field?

• The integers Zcorrect
• The rationals Q
• The reals R
• The complex numbers C

The integers Z are not a field because most integers (like 2) have no integer multiplicative inverse; Q, R, and C are fields.

19. Build a true statement about the empty set.

Answer: The empty set has no elements

The empty set, written {}, contains nothing at all, so it has no elements.

20. Which statements about the empty set are true? (Select all that apply.)

• It has 0 elementscorrect
• It is a subset of every setcorrect
• It contains the number 0
• It is equal to {0}

The empty set has 0 elements and is a subset of every set; it does not contain 0 and is not equal to {0}.

21. A function that is both injective (one-to-one) and surjective (onto) is called ____.

• bijectivecorrect
• constant
• continuous
• linear

A function that is both one-to-one and onto is a bijection, or bijective.

22. Match each number system to the symbol used for it.

Answer: Integers = Z; Rationals = Q; Reals = R

Z is the standard symbol for integers, Q for rationals (quotients), and R for the real numbers.

23. The intersection of two disjoint sets is the empty set.

Answer: True

True. Disjoint means the sets share no elements, so their intersection contains nothing.

24. How many elements are in the power set of a set with 4 elements? (That is 2^4.)

Answer: 16

The power set of an n-element set has 2^n members, so 2^4 = 16.

25. Which statements about the identity matrix I are true? (Select all that apply.)

• I times A equals Acorrect
• The determinant of I is 1correct
• I has 0s on its main diagonal
• I is not a square matrix

I leaves any matrix unchanged (I*A = A) and has determinant 1; it does have 1s on its diagonal and is square.

26. The ____ of the 2x2 matrix [[a,b],[c,d]] equals ad - bc.

• determinantcorrect
• trace
• inverse
• transpose

That number is the determinant; it tells you whether the matrix is invertible.

27. Match each matrix or vector expression to its value.

Answer: det [[1,0],[0,1]] = 1; trace [[2,0],[0,3]] = 5; (1, 2) + (3, 4) = (4, 6)

det [[1,0],[0,1]] = 1, the trace (diagonal sum) of [[2,0],[0,3]] = 5, and (1,2) + (3,4) adds entrywise to (4,6).

28. What is the determinant of the 2x2 matrix [[1,2],[3,4]]?

• -2correct
• 2
• 10
• -10

For [[a,b],[c,d]] the determinant is ad - bc, so here 1*4 - 2*3 = 4 - 6 = -2.

29. Three of these are matrix operations. Which one is NOT?

• Integrationcorrect
• Transpose
• Determinant
• Inverse

Integration is a calculus operation; transpose, determinant, and inverse all act on matrices.

30. Order these vectors from smallest to largest magnitude (length).

Answer: (1, 0) -> (0, 2) -> (3, 4) -> (6, 8)

The magnitudes are |(1,0)| = 1, |(0,2)| = 2, |(3,4)| = 5, and |(6,8)| = 10.