Set Theory Basics in Mathematics

A set is a collection of distinct objects or elements that are well-differentiated.

A subset is a set whose elements are all contained within another set, while a superset contains all the elements of a subset and possibly more.

Cardinality refers to the number of elements in a set, which helps classify and compare sets.

<p>The union of two sets combines all unique elements from both sets into a new set.</p> Signup and view all the answers

<p>A number splits another if it divides it evenly, leaving no remainder.</p> Signup and view all the answers

<p>An equivalence relation is a binary relation that is reflexive, symmetric, and transitive.</p> Signup and view all the answers

<p>A power set is the set of all possible subsets of a given set, including the empty set and the set itself.</p> Signup and view all the answers

<p>The intersection of sets contains only those elements that are present in both sets.</p> Signup and view all the answers

<p>A set is a superset of its subsets; for example, if P is the set of all persons and P' is the set of persons with blond hair, then P' ⊆ P.</p> Signup and view all the answers

<p>The notation N ⊈ M indicates that set N is not a subset of set M.</p> Signup and view all the answers

<p>Two sets M and N are equal if M ⊆ N and N ⊆ M, such as M = {1, 2, 3} and N = {1, 2, 3}, where both conditions hold.</p> Signup and view all the answers

<p>The union of two sets M and N, denoted as M ∪ N, includes all elements that are in M or N, without duplicates.</p> Signup and view all the answers

<p>The intersection of two sets M and N, denoted as M ∩ N, consists of elements that are common to both sets.</p> Signup and view all the answers

<p>The difference between sets M and N, denoted M \ N, includes elements that are in M but not in N.</p> Signup and view all the answers

<p>Yes, a set is always a subset of itself, denoted as M ⊆ M.</p> Signup and view all the answers

<p>The set of all odd natural numbers N' is a subset of the set of natural numbers ℕ, so N' ⊆ ℕ.</p> Signup and view all the answers

<p>It means there exists an integer k such that $a + b = k \cdot n$.</p> Signup and view all the answers

<p>We write it as $a \equiv b \mod m$.</p> Signup and view all the answers

<p>Reflexivity, symmetry, and transitivity.</p> Signup and view all the answers

<p>Since $20 - 35 = -15$ and $-15 = -3 \cdot 5$, it follows that 5 divides the difference.</p> Signup and view all the answers

<p>It means that $m$ divides $(b - a)$, or that $b - a = k \cdot m$ for some integer k.</p> Signup and view all the answers

<p>Reflexivity means that for any integer a, $a \sim_n a$ because $n | (a - a)$, which equals 0.</p> Signup and view all the answers

<p>Symmetry implies that if $a \sim_n b$, then $b \sim_n a$, since if $n | (a - b)$, then $n | (b - a)$.</p> Signup and view all the answers

<p>Because $21 - 0 = 21$ and $7 | 21$ since $21 = 3 \cdot 7$.</p> Signup and view all the answers

<p>We conclude that if a and b are in the same equivalence class, then n divides b - a.</p> Signup and view all the answers

<p>ℤ can be decomposed into equivalence classes represented by {0 + kn, 1 + kn, ..., (n-1) + kn} for k ∈ ℤ.</p> Signup and view all the answers

<p>We can say they are equal because n ∈ [0] and n ∈ [n], meaning both classes contain the same elements based on their definitions.</p> Signup and view all the answers

<p>It can be inferred that [kn] = [n] for any integer k, as both classes consist of the same elements derived from their definitions.</p> Signup and view all the answers

<p>The equation implies that the difference b - a is a multiple of n, demonstrating that a and b belong to the same equivalence class.</p> Signup and view all the answers

<p>An equivalence class [a] under the relation ∼n consists of all integers of the form a + kn for k ∈ ℤ.</p> Signup and view all the answers

<p>The set [a] is not empty because it contains at least the element a itself, as a = a + 0n.</p> Signup and view all the answers

<p>If x, y ∈ [a], then x = a + kn and y = a + k'n, and thus x - y = n(k - k'), implying n|(x - y) and x ∼n y.</p> Signup and view all the answers

<p>The main theorem states that every element x in a set M is contained in exactly one equivalence class with respect to the equivalence relation R.</p> Signup and view all the answers

<p>Reflexivity ensures that x ∼ x, thus confirming that x is in Lx, making Lx non-empty.</p> Signup and view all the answers

<p>Symmetry allows us to assert y' ∼ x if y ∼ x, and transitivity ensures that if y ∼ x and x ∼ y', then y ∼ y'.</p> Signup and view all the answers

<p>This implies that Lx is a subset of L'x, demonstrating that both equivalence classes must be identical.</p> Signup and view all the answers

<p>The integer n determines the grouping of integers into equivalence classes based on the divisibility of their differences by n.</p> Signup and view all the answers

<p>Equivalence classes L and L' are either equal or disjoint.</p> Signup and view all the answers

<p>It means that the set can be split into distinct equivalence classes where each class groups elements that are equivalent under the relation.</p> Signup and view all the answers

<p>Because all elements of an equivalence class are equivalent and share the same property defined by the equivalence relation.</p> Signup and view all the answers

<p>The equivalence classes are represented by SM for medicine, SI for computer science, and SG for history.</p> Signup and view all the answers

<p>The uniqueness of equivalence classes ensures that any two classes either share all their elements (and are thus the same) or have no elements in common (are disjoint).</p> Signup and view all the answers

<p>By showing that they contain at least one common element, which confirms they are equivalent classes.</p> Signup and view all the answers

<p>Yes, equivalence classes can contain multiple representatives because all elements in a class are considered equivalent.</p> Signup and view all the answers

<p>It means that they share no common elements and therefore are distinct classes.</p> Signup and view all the answers

A collection of distinct objects, often called elements, that are clearly distinguishable from each other. These objects can be real-world things or mathematical constructs.

A set containing only elements that are also members of another set.

A set that contains all the elements of another set, possibly along with additional elements.

The combination of two sets, including all elements from both.

The set containing only the elements present in both sets.

The set containing elements that are present in the first set but not in the second set.

The number of elements in a set.

The set of all possible subsets of a given set, including the empty set and the set itself.

Two sets are equal if they contain the same elements, regardless of order.

A symbol used to indicate that a set is a subset of another set.

A symbol used to indicate that a set is not a subset of another set.

A relation where each element is related to itself, and if element a is related to element b, then b is also related to a, and if a is related to b and b is related to c, then a is related to c.

A set of elements that are all equivalent to each other under a given equivalence relation. All elements within a class share a common property denoted by the relation.

The set of all integers that can be obtained by adding a multiple of n to a fixed integer.

The process of dividing a set into distinct, non-overlapping subsets, where each element belongs to exactly one subset. Each subset contains only elements that are equivalent to each other.

Two sets are disjoint if they have no elements in common. In the context of decomposition, this ensures that each element belongs to only one equivalence class.

If a number 'n' divides another number 'a', we write 'n|a'. This means 'a' can be divided by 'n' without leaving a remainder.

If 'n' divides both 'a' and 'b', then it also divides their sum (a + b) and their difference (a - b).

Two integers 'a' and 'b' are congruent modulo 'm' if their difference (b - a) is divisible by 'm'. This means they have the same remainder when divided by 'm'.

The relationship 'a ∼n b', where 'a' and 'b' are integers and 'n' is a natural number, defines an equivalence relation on the set of integers (ℤ).

Reflexivity: For any integer 'a', 'a' is congruent to itself modulo 'n' (a ∼n a) because 'n' divides (a - a) = 0.

Symmetry: If 'a' is congruent to 'b' modulo 'n' (a ∼n b), then 'b' is also congruent to 'a' modulo 'n' (b ∼n a)

Transitivity: If 'a' is congruent to 'b' modulo 'n' (a ∼n b) and 'b' is congruent to 'c' modulo 'n' (b ∼n c), then 'a' is also congruent to 'c' modulo 'n' (a ∼n c).

The set of all equivalence classes of M with respect to R, where each element of M belongs to exactly one equivalence class.

Every element of a set belongs to exactly one equivalence class defined by a relation.

The set of all elements related to a specific element 'a' with respect to the relation. Formally: Lx = {y ∈ M | y ∼ x}.

A relation on integers where two integers are related if their difference is a multiple of 'n'. Written as a ∼n b, meaning a - b is divisible by n.

The set of all integers of the form a + kn, where 'k' is any integer. It represents the equivalence class of 'a' with respect to the relation ∼n.

Given an equivalence relation R, it proves that Lx (the equivalence class of x) is an equivalence class and that every element of the set belongs to a unique equivalence class.

Two equivalence classes are either identical or have no elements in common.

A decomposition of a set into disjoint equivalence classes, meaning the classes do not overlap and cover all the elements of the original set.

Any element within an equivalence class can be considered as a representative of that class.

The process of selecting an arbitrary representative from an equivalence class, where any member of the class can be chosen because they are all equivalent based on the defining property of the class.

The property that all elements in an equivalence class share based on which the equivalence relation is defined. For example, the property 'the students all study the same subject' could be the property shared by the elements in an equivalence class of students.