sets class 11

A set is a collection of distinct objects, considered as an object in its own right.

Sets are denoted by capital letters, and elements are enclosed in curly brackets. Example: A = {1, 2, 3}.

An element is an individual object within a set.

The symbol ∈ denotes that an element belongs to a set.

The symbol ∉ denotes that an element does not belong to a set.

The null set, or empty set, is a set with no elements, denoted by ∅ or { }.

The universal set contains all objects under consideration, typically denoted by U.

A subset is a set where every element is also in another set. Denoted by ⊆.

A proper subset is a subset that is not equal to the original set. Denoted by ⊂.

The power set of a set S is the set of all possible subsets of S. Denoted by P(S).

The union of sets A and B is the set of elements that are in A, in B, or in both. Denoted by A ∪ B.

The intersection of sets A and B is the set of elements that are in both A and B. Denoted by A ∩ B.

The difference of sets A and B is the set of elements that are in A but not in B. Denoted by A - B.

The complement of a set A refers to elements in the universal set that are not in A. Denoted by A'.

Disjoint sets are sets that have no elements in common. Their intersection is the null set.

A Venn diagram is a visual tool used to represent sets and their relationships using overlapping circles.

The cardinality of a set is the number of elements in the set. Denoted by |A|.

A finite set has a countable number of elements, while an infinite set has an uncountable number of elements.

The principle of inclusion-exclusion is used to calculate the size of the union of multiple sets by including and excluding the sizes of intersections.

The Cartesian product of sets A and B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. Denoted by A × B.

A ∪ A = A

A ∩ A = A

A ∪ B = B ∪ A

A ∩ B = B ∩ A

(A ∪ B) ∪ C = A ∪ (B ∪ C)

(A ∩ B) ∩ C = A ∩ (B ∩ C)

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

(A ∪ B)' = A' ∩ B'

(A ∩ B)' = A' ∪ B'

(A')' = A

A ∪ (A ∩ B) = A

A ∩ (A ∪ B) = A

The symmetric difference of sets A and B is the set of elements in either A or B but not in both. Denoted by A △ B.

The complement of the universal set U is the null set, ∅.

A set with a specific number of elements.

A set with an unlimited number of elements.

A set with exactly one element.

Every set is a subset of the universal set U.

A ∪ U = U

A ∩ U = A

A ∪ ∅ = A

A ∩ ∅ = ∅

Set-builder notation defines a set by a property that its members must satisfy. Example: {x | x is a natural number less than 5}.

In a Venn diagram, the complement of a set A is the region outside A within the universal set U.

The complement of A ∩ B is given by (A ∩ B)' = A' ∪ B'.

The complement of A ∪ B is given by (A ∪ B)' = A' ∩ B'.
Got it! Here are additional flashcards focusing strictly on formulas and properties related to sets:

|A ∪ B| = |A| + |B| - |A ∩ B|

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |C ∩ A| + |A ∩ B ∩ C|

The number of subsets is 2^n.

The number of elements in the power set is 2^n.

|A'| = |U| - |A|

|A △ B| = |A ∪ B| - |A ∩ B| = |A| + |B| - 2|A ∩ B|

A ∩ B = (A' ∪ B')'

A ∪ B = (A' ∩ B')'

|A × B| = |A| × |B|

|A × B × C| = |A| × |B| × |C|

(A ∪ B)' = A' ∩ B'

(A ∩ B)' = A' ∪ B'

For n sets: |A1 ∪ A2 ∪ ... ∪ An| = Σ |Ai| - Σ |Ai ∩ Aj| + Σ |Ai ∩ Aj ∩ Ak| - ... + (-1)^(n+1) |A1 ∩ A2 ∩ ... ∩ An|

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |C ∩ A| + |A ∩ B ∩ C|

The number of proper subsets is 2^n - 1.

For finite sets A and B, the number of ordered pairs in A × B is |A| × |B|.

The number of elements in the power set is 2^k.

For n sets: |A1 ∩ A2 ∩ ... ∩ An| = |A1| + |A2| + ... + |An| - Σ |Ai ∪ Aj| + Σ |Ai ∪ Aj ∪ Ak| - ... + (-1)^(n+1) |A1 ∩ A2 ∩ ... ∩ An|