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CMSC250 SECTIONS 0303 & 0304 MIDTERM REVIEW Sri Kankanahalli Discussion 10: 9 March 2016 Office Hrs: Mon. and Wed. 4-6PM AVW 1112 Topics for Today • Midterm 1 is March 21st ! ! ! • Review: • Propositional logic • Binary arithmetic (1.1, 1.2, 1.4) (Feb 3. slides 18-21) • Two’s complement • Proof techniques • Set theory • Functions (1.7, a bit of 1.8) (2.1, 2.2) (2.3) • Floor and ceiling functions • Number theory (4.1) Propositional Logic • Know: • Logical connectives (including implication and biimplication) • Quantifiers (∀, ∃) • Translating from English to propositional logic, in both directions • A(X): X is a cow. • B(X): X is blue. • “All cows are blue.” • ∀ x [A(x) → B(x)] • ∃ x ¬A(x) ∧ B(x) • “There exists a thing that is not a cow and is blue.” Propositional Logic • Don’t need to know (for this test): • System specifications • Circuits Propositional Logic • Slides: • Feb. 1, everything • Feb. 8, material on quantifiers • Feb. 10, everything • Practice problems: • Translating from English to propositional logic • Chapter 1.1, #10-12 • Working with quantifiers • Chapter 1.4, #10, 32, 33 Binary Arithmetic • Know: • How to convert numbers from base 10 to binary • Two’s complement arithmetic • Don’t need to know • How to convert to other bases (octal, hexadecimal, etc.) • Though this is still a good skill to have in life! Binary Arithmetic • Slides: • Feb. 3, material on binary arithmetic • Practice problems: • Convert 59 to an 8-bit two’s complement binary number. • Convert -63 to an 8-bit two’s complement binary number. • Express (21 – 95) as an 8-bit two’s complement binary number. Proof Techniques • Know: • All our basic methods of proof • Direct proof • Proof by contraposition • Proof by contradiction • General proof techniques • Constructing a counterexample • Proof by cases Proof Techniques • Slides: • Feb. 17, everything • Practice problems: • Basic proof methods • Chapter 1.7, #1-5, 6, 13 • Proof by counterexample and/or cases • Chapter 1.8, #3, 6 Set Theory • Know: • Set operations (union, intersection, difference) • Definition of subset and proper subset • Proving properties of sets • Proof by “element chasing” • Proof by derivation • Don’t need to know (for this test): • Set identities (you’ll get a sheet, like the one you had on the HW) Set Theory • Slides: • Feb. 22, everything • Practice problems: • “Element chasing” proofs • Chapter 2.2, #16, 19 • Derivational proofs • Chapter 2.2, #17, 18 • Prove (A – B) ∩ (B – A) = ∅, both ways. Functions • Know: • Finding the domain and codomain of a function • Injectivity, surjectivity, bijectivity – and how to prove them • How to take the inverse of a function, and verify it • Floor and ceiling functions • Don’t need to know (for this test): • Partial functions • Binary relations Functions • Slides: • Feb. 24, everything • Feb. 29, material on floor/ceiling functions • Practice problems: • Finding domain and codomain • Chapter 2.3, #7 • Determining injectivity/surjectivity/bijectivity • Chapter 2.3, #12-15, 22, 23 • Taking inverses of functions • Find the inverse of f(x) = 3x + 4 + 5/x • Floor and ceiling functions • Chapter 2.3, #8, 9 Number Theory • Know: • Proving things about even/odd numbers • Divisibility • Modular arithmetic • Proving statements like: • “If n is odd, n2 ≡ 1 (mod 8).” • “If n is odd and m ≡ 3 (mod 4), then (n2 + m) is divisible by 4.” (More complicated than midterm.) • Proving small roots are irrational • Using modular arithmetic • Using the Unique Factorization Theorem (slides later!) Number Theory • Don’t need to know (for this test): • The division algorithm • Modular exponentiation • Proofs about primes Number Theory • Slides: • Feb. 29, number theory and divisibility • Mar. 7, everything • Practice problems: • Proving things about even/odd numbers • Chapter 1.7, #1-5, 6, 13 • Divisibility • Chapter 4.1, #5-8 • Modular arithmetic • Chapter 4.1, #38-40 • Proving irrationality • Prove √3 is irrational, once with modular arithmetic, and once with the Unique Factorization Theorem. Unique Factorization Theorem • Also called the “Fundamental Theorem of Arithmetic” • Theorem: “Every integer can be expressed as a product of unique prime numbers.” • 24 • 160 • x =3*2*2*2 = 3 * 23 =5*4*4*2 = 5 * 42 * 2 = p1a1 * p2a2 * … * pnan Unique Factorization Theorem • Proof: √2 is irrational. • A proof by contradiction (like usual): Assume √2 is rational. Then √2 = a / b, for a and b with no common factors. • So 2 = a2 / b2. • So a2 = 2b2. • We’ve done this many times before. Only the next part differs. Unique Factorization Theorem • Proof: √2 is irrational. • So a2 = 2b2. • By the UFT, we can write a and b as a unique product of prime factors. • a = p1x1 * p2x2 * … * pnxn • b = q1y1 * q2y2 * … * qnyn • So, we can write a2 and b2 as: • a2 = p12x1 * p22x2 * … * pn2xn • b2 = q12y1 * q22y2 * … * qn2yn Unique Factorization Theorem • Proof: √2 is irrational. • a2 = 2b2. • So, we can write a2 and b2 as: • a2 = p12x1 * p22x2 * … * pn2xn • b2 = q12y1 * q22y2 * … * qn2yn • We see a2 and b2 have all even powers, for each prime in their factorizations. • So, a2 and b2 would both have an even number of 2s in their factorizations. • So, 2b2 would have an odd number of 2s. • Since 2b2 has an odd number of 2s in its factorization, and a2 has an even number of 2s, by the UFT they can’t be equal! Contradiction. Unique Factorization Theorem • Proof: √2 is irrational. • a2 = 2b2. • Since 2b2 has an odd number of 2s in its factorization, and a2 has an even number of 2s, by the UFT they can’t be equal! Contradiction. • Because assuming that √2 is rational leads to a contradiction, √2 must be irrational. QED