Discrete Mathematics

Theorem 2.18

(1) Commutative Laws (a) P ∨ Q ≡ Q ∨ P (b) P ∧ Q ≡ Q ∧ P (2) Associative Laws (a) P ∨ (Q ∨ R) ≡ (P ∨ Q) ∨ R (b) P ∧ (Q ∧ R) ≡ (P ∧ Q) ∧ R (3) Distributive Laws (a) P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R) (b) P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R) (4) De Morgan's Laws (a) ∼ (P ∨ Q) ≡ (∼P) ∧ (∼Q) (b) ∼ (P ∧ Q) ≡ (∼P) ∨ (∼Q).

Notes on sets and set operations...

(A ⋂ B) ⊆ (A ⋃ B) Given D = { 1, { 1 }, 2, 3 } { 1 } ∈ D (this element refers to the second element in the set) { 1 } ⊆ D (this subset refers to the first item shown in D. When using union and intersection, note that the two are commutative operations.

Let A = {0,{0},{0,{0}}}. (a) Determine which of the following are elements of A: 0, {0}, {{0}}. (b) Determine |A|. 24 Chapter 1 Sets (c) Determine which of the following are subsets of A: 0, {0}, {{0}}. For (d)-(i), determine the indicated sets. (d) {0} ∩ A (e) {{0}} ∩ A (f) {{{0}}} ∩ A (g) {0} ∪ A (h) {{0}} ∪ A (i) {{{0}}} ∪ A.

(a) While 0 and {0} are elements of A, {{0}} is not an element of A. (b) The set A has three elements: 0, {0}, {0, {0}}. Therefore, |A| = 3. (c) The integer 0 is not a set and so cannot be a subset of A (or a subset of any other set). Since 0 ∈ A and {0} ∈ A, it follows that {0} ⊆ A and {{0}} ⊆ A. (d) Since 0 is the only element that belongs to both {0} and A, it follows that {0} ∩ A = {0}. (e) Since {0} is the only element that belongs to both {{0}} and A, it follows that {{0}} ∩ A = {{0}}. (f) Since {{0}} is not an element of A, it follows that {{{0}}} and A are disjoint sets and so {{{0}}} ∩ A = ∅. (g) Since 0 ∈ A, it follows that {0} ∪ A = A. (h) Since {0} ∈ A, it follows that {{0}} ∪ A = A. (i) Since {{0}} ∈/ A, it follows that {{{0}}} ∪ A = {0,{0},{{0}},{0,{0}}}.

Earmark

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1. How would you express the complement of A in terms of the universal set using set builder notation? 2. If U = Z, what is N' ? 3. If U = R, what is Q' ?

1. A' = U - A = {x | x ∈ U and x ∉ A} 2. {0, -1, -2, ...} 3. I

1. Theorem 2.17: Stating an implication as a disjunction 2. Theorem 2.21a: Stating the negation of the implication as a conjunction.

1. P ⇒ Q ≡ (∼P) ∨ Q 2. Commutative Laws (a): P ∨ Q ≡ Q ∨ P

Find two sets A and B such that A is both an element of and a subset of B.

A = { 1 } B = { 1, {1} } It is important to note that the second element of B arises from needing A to be an element of B, whereas the first element arises from the requirement that A is a subset of B.

Example 1.13 Express the following intervals in set builder notation. A = [-3,3] B = (-∞,-2)⋃(2,∞) C = [-3,5] Find the following: 1. A⋂B 2. A - B 3. B⋂C 4. B⋃C 5. B - C 6. C - B

A = {x ∈ R : |x| ≤ 3} B = {x ∈ R : |x| > 2} C = {x ∈ R : |x - 1| ≤ 4} 1. [-3, -2) ⋃ (2, 3] 2. [-2, 2] 3. [-3, -2)⋃(2, 5] 4. (-∞,∞) 5. (-∞, -3)⋃(5, ∞) 6. [-2, -2]

Pairwise Disjoint

A collection S of subsets of a set A is called pairwise disjoint if every two distinct subsets that belong to S are disjoint. Note that the intersection of a set may be the null set, but it is the pairs of subsets that determine whether or not the set is pairwise disjoint.

Corollary and Lemma

A corollary is a mathematical result that can be deduced from, and is thereby a consequence of, some earlier result. A lemma (German hilfsatz, meaning 'helping theorem') is a mathematical result that is useful in establishing the truth of some other result.

Proper Subsets and Power Sets

A set A is a proper subset of B if A ⊆ B but A ≠ B. If A is a proper subset, we write A ⊂ B. The set consisting of all subsets of a given set A is called the power set of A and is denoted by P(A). Notice that for each set, we have | P(A) = 2^|A| |. In fact, if A is any finite set, with n elements say, then P(A) has 2^n elements- That is, | P(A) | = 2^|A| for every finite set A. In simpler terms, the cardinality of the power set of A is two raised to the cardinality of the set A. Note that the P(A) of a set A = {{}} = Ø. Also, note that |A| = 0, and 2^0 = 1, making the power set of a set that only contains the null set equal to 1.

Subsets

A set A is called a subset of a set B if every element of A also belongs to B. Note that, if x ∈ A, and B ⊆ C, x ∈ C. Every element is a subset of itself. If B is not a subset C, there must be some element of B that is not an element of C.

Name four ways to write the set of even numbers in set-builder notation.

E = {y | y is an even integer} E = {2x | x is an integer} E = {y | y = 2x for some x ∈ Z} E = {2x | x ∈ Z}

Theorem 3.16 Let x, y ∈ Z. Then x and y are of the same parity if and only if x + y is even.

First, assume that x and y are of the same parity. We consider two cases. Case 1. x and y are even. Then x = 2a and y = 2b for some integers a and b. So x + y = 2a + 2b = 2(a + b). Since a + b ∈ Z, the integer x + y is even. Case 2. x and y are odd. Then x = 2a + 1 and y = 2b + 1, where a, b ∈ Z. Therefore, x + y = (2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1). Since a + b + 1 is an integer, x + y is even. 3.4 For the converse, assume that x and y are of opposite parity. Again, we consider two cases. Case 1. x is even and y is odd. Then x = 2a and y = 2b + 1, where a, b ∈ Z. Then x + y = 2a + (2b + 1) = 2(a + b) + 1. Since a + b ∈ Z, the integer x + y is odd. Case 2. x is odd and y is even. The proof is similar to the proof of the preceding case and is therefore omitted.

Example 2.8

For a triangle T , let P(T ) : T is equilateral. and Q(T ) : T is isosceles. Thus, P(T ) and Q(T ) are open sentences over the domain S of all triangles. Consider the implication P(T ) ⇒ Q(T ), where the domain then of the variable T is the set S. For an equilateral triangle T1, both P(T1) and Q(T1) are true statements and so P(T1) ⇒ Q(T1) is a true statement as well. If T2 is not an equilateral triangle, then P(T2) is a false statement and so P(T2) ⇒ Q(T2) is true. Therefore, P(T ) ⇒ Q(T ) is a true statement for all T ∈ S. We now state P(T ) ⇒ Q(T ) in a variety of ways: - If T is an equilateral triangle, then T is isosceles. - A triangle T is isosceles if T is equilateral. - A triangle T being equilateral implies that T is isosceles. - A triangle T is equilateral only if T is isosceles. - For a triangle T to be isosceles, it is sufficient that T be equilateral. - For a triangle T to be equilateral, it is necessary that T be isosceles.

The set of open and closed intervals

For a, b are elements of R and a < b, the open interval (a, b) is the set: (a, b) = {x ∈ R | a < x < b} For a, b are elements of R and a < b, the closed interval [a, b] is the set: [a, b] = {x ∈ R | a ≤ x ≤ b}

Example 1.7 Let S = {1, {2}, {1, 2}}. I. Determine the elements of S: 1, {1}, 2, {2}, {1, 2}, {{1, 2}} II. Determine which of the following are subsets of S: 1, {2}, {1, 2}, {{1},2}, {1, {2}}, {{1}, {2}}, {{1, 2}}

I. 1, {2}, {1, 2} II. 1, {1, {2}}, {{1, 2}}

Necessary and Sufficient

If is sufficient, then is necessary.

Notes on Sets I

If the set T = {0, {1, 2, 3}, 4, 5} The set T has four elements, but 2 ∉ T. It is an element of the set {1, 2, 3}, which is an element of T.

Disjoint

If two sets A and B have no elements in common, then A ⋂ B = Ø and A and B are said to be disjoint. For example, Q and I are disjoint.

Trivial Proof

If, in a given implication, Q is always true, a proof that demonstrates this is called a trivial proof.

Binary operations

In mathematics, a binary operation or dyadic operation is a calculation that combines two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation on a set is an operation whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, multiplication. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication and conjugation in groups. An operation of arity two that involves several sets is sometimes also called a binary operation. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar. Such binary operations may be called simply binary functions. Binary operations are the keystone of most algebraic structures, that are studied in algebra, in particular in semigroups, monoids, groups, rings, fields, and vector spaces.

Unary Operation

In mathematics, a unary operation is an operation with only one operand, i.e. a single input.[1] This is in contrast to binary operations, which use two operands.[2] An example is the function f : A → A, where A is a set. The function f is a unary operation on A. Common notations are prefix notation (e.g. +, −, ¬), postfix notation (e.g. factorial n!), functional notation (e.g. sin x or sin(x)), and superscripts (e.g. transpose AT). Other notations exist as well. For example, in the case of the square root, a horizontal bar extending the square root sign over the argument can indicate the extent of the argument.

Other set of numbers

Irrational numbers are real numbers that are not rational, like the square roots of 2 and 3, π, or e. often represented by I. Complex numbers are numbers of the form a + bi where a, b are elements of R and i = sqrt(-1). A complext number where b = 0 can be expressed as only a. Accordingly, a + 0i is a real number. Thus every real number is a complex number. Complex numbers are often represented by C. Be aware that 0 is not listed as a natural number in the textbook (it is, however, listed as an integer).

If K = {x ∈ C | x^2 +1 = 0}, what is K?

K = {-i, i}

Vacuous Proof

Let P(x) and Q(x) be open sentences over a domain S. Then ∀x ∈ S, P(x) ⇒ Q(x) is a true statement if it can be shown that P(x) is false for all x ∈ S (regardless of the truth value of Q(x)), according to the truth table for implication. Such a proof is called a vacuous proof of ∀x ∈ S, P(x) ⇒ Q(x). T

Three ways to list sets...

List them, use Set-builder notation, and reserved sets. It is common to list the sets as uppercase letters (as in A, B, C, S, X, Y) and the elements as lowercase letters.

Number sets in notation

N ⊆ Z, and Q ⊆ R. In addition, R ⊆ C. Since Q ⊆ R and R ⊆ C, it therefore follows that Q ⊆ C. N ⋂ Z = N and Q ⋂ R = Q. Furthermore, R - Q = I. Q and I are said to be disjoint.

Name the set of odd integers in set-builder notation.

O = {2k + 1 : k∈Z} O = {..., -5, -3, -1, 1, 3, 5, ...}

In which of the following sets is the integer -2 an element? S1 = {-1, -2, {-1}, {-2}, {-1, -2}} S2 = {x ∈ N | -x ∈ N} S3 = {x ∈ Z | x^2 = 2^x} S4 = {x ∈ Z | |x| = -x} S5 = {{-1, -2}, {-2, -3}, {-1, -3}}

Only S1 and S4. Observe that S2 = {} and S5 has only sets (not integers).

Original, Converse, Contrapositive, Inverse

Original: P ⇒ Q Converse: Q ⇒ P Contrapositive: ~Q ⇒ ~P Inverse: ~P ⇒ ~Q Note that the original conditional and its contrapositive are logically equivalent, whereas the converse and the inverse are logically equivalent.

Example 2.7

P1(x) : x = −3. and P2(x) : |x| = 3, where x ∈ R, that is, where the domain of x is R in each case. We can then form the following open sentences: ∼ P1(x) : x = −3. P1(x) ∨ P2(x) : x = −3 or |x| = 3. P1(x) ∧ P2(x) : x = −3 and |x| = 3. P1(x) ⇒ P2(x) : If x = −3, then |x| = 3.

Theorem 3.12 Let x ∈ Z. Then x^2 is even if and only if x is even. This theorem can be restated as 'The square of every even integer is even.' This example is on page 87. In addition there is an exercise requiring a similar proof. Consider also the following statement: Let x ∈ Z. Then x^2 is odd if and only if x is odd. This is a rephrasing of the theorem.

Proof: Assume that x is even. Then x = 2a for some integer a. Therefore, x^2 = (2a)^2 = 4a^2 = 2(2a^2 ). Because 2a^2 ∈ Z, the integer x^2 is even. For the converse, assume that x is odd. So x = 2b + 1, where b ∈ Z. Then x^2 = (2b + 1)^2 = 4b^2 + 4b + 1 = 2(2b^2 + 2b) + 1. Since 2b^2 + 2b is an integer, x^2 is odd.

Theorem 3.17 Let a and b be integers. Then ab is even if and only if a is even or b is even.

Proof: First, assume that a is even or b is even. Without loss of generality, let a be even. Then a = 2x for some integer x. Thus ab = (2x)b = 2(xb). Since xb is an integer, ab is even. For the converse, assume that a is odd and b is odd. Then a = 2x + 1 and b = 2y + 1, where x, y ∈ Z. Hence ab = (2x + 1)(2y + 1) = 4x y + 2x + 2y + 1 = 2(2x y + x + y) + 1. Since 2x y + x + y is an integer, ab is odd.

Name three ways to write the set of squares of integers in set-builder notation.

S = {x^2 | x is an integer} S = {x^2 | x ∈ Z} S = {0, 1, 4, 9, ...}

Index Set

See image.

Indexed collections of Sets

See image.

Special Sets of Numbers

See image. The set of positive real numbers is denoted by R[superscript]+. The real numbers that can be expressed in the form m/n, where m, n ∈ Z and n ≠ 0. Those that cannot are irrational. Note that sqrt(2), sqrt(3), cbrt(2), pi, and e are irrational. That is, none of them can be expressed as the ratio of two integers. That is, if x ∈ R and x ∉ Q, then x ∈ I.

Example 2.25

Suppose that we are considering the set A = {1, 2, 3} and its power set P(A), the set of all subsets of A. Then the quantified statement For every set B ∈ P(A), A − B = ∅. (2.6) is false since for the subset B = A = {1, 2, 3}, we have A − B = ∅. The negation of the statement (2.6) is There exists B ∈ P(A) such that A − B = ∅. (2.7) The statement (2.7) is therefore true since for B = A ∈ P(A), we have A − B = ∅. The statement (2.6) can also be written as If B ⊆ A, then A − B = ∅. (2.8) Consequently, the negation of (2.8) can be expressed as There exists some subset B of A such that A − B = ∅.

Suppose we are asked to verify that ∼ (P ⇒ Q) ≡ P ∧ (∼Q) for every two statements P and Q...

Suppose we are asked to verify that ∼ (P ⇒ Q) ≡ P ∧ (∼Q) for every two statements P and Q. Using the logical equivalence of P ⇒ Q and (∼P) ∨ Q from Theorem 2.17 and Theorem 2.18(4a), we see that ∼ (P ⇒ Q) ≡ ∼((∼P) ∨ Q) ≡ (∼(∼P)) ∧ (∼Q) ≡ P ∧ (∼Q)

The difference of two sets

The difference A - B of two sets A and B (also written as A \ B ) is defined as A - B = {x : x ∈ A and x ∉ B }. For example, R - Q = I

Cardinality

The number of elements in a set. Note that the cardinality |S| of a set S includes a null set, if there is one as an element, but that the null set itself has a cardinality of 0.

Relative Complement of sets

The set difference A - B is sometimes called the relative complement of B in A. The definition: A - B = { x | x ∈ A and x ∉ B} This can also be expressed in terms of complements, namely, A - B = A ⋂ B'

Theorem 2.21 Example 2.20 (the negation of an implication and a biconditional) is repeated in the following theorem.

Theorem 2.21 For statements P and Q, (a) ∼ (P ⇒ Q) ≡ P ∧ (∼Q) (b) ∼ (P ⇔ Q) ≡ (P ∧ (∼Q)) ∨ (Q ∧ (∼P)).

Theorem 3.9

Theorem 3.9 For every two statements P and Q, the implication P ⇒ Q and its contrapositive are logically equivalent; that is, P ⇒ Q ≡ (∼Q) ⇒ (∼P).

Inclusion Exclusion Principle

This is a combinatoric and computer scientific tool- see image.

Parity

Two integers x and y are said to be of the same parity if x and y are both even or are both odd. The integers x and y are of opposite parity if one of x and y is even and the other is odd. For example, 5 and 13 are of the same parity, while 8 and 11 are of opposite parity. Because the definition of two integers having the same (or opposite) parity requires the two integers to satisfy one of two properties, any result containing these terms is likely to be proved by cases. The following theorem presents a characterization of two integers that are of the same parity

Example 2.20

Using the second of De Morgan's Laws and (2.1), we can establish a useful logically equivalent form of the negation of P ⇔ Q by the following string of logical equivalences: ∼(P ⇔ Q) ≡ ∼((P ⇒ Q) ∧ (Q ⇒ P)) ≡ (∼(P ⇒ Q)) ∨ (∼(Q ⇒ P)) ≡ (P ∧ (∼Q)) ∨ (Q ∧ (∼P)).

Tautology and Contradiction

When a truth table results entirely in T value, that particular logical statement is called a tautology. Example: P U ~P. When a truth table results entirely in F values, that particular statement is called a contradiction.

Textbook

file:///C:/Users/geoff/OneDrive/Desktop/Spring%202021/Discrete%20Mathematics/Mathematical%20Proofs%20-%203rd%20Edition%20-%20Chartrand.pdf

Set symbols

https://www.rapidtables.com/math/symbols/Set_Symbols.html#:~:text=Table%20of%20set%20theory%20symbols%20%20%20,{9,66}%20⊄%20{9,14,28}%20%2022%20more%20rows

What is the order of the reserved sets?

ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ ⊆ ℂ