Sets in mathematics
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Transcript of Sets in mathematics
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MATHEMATICS PRESENTATION PRESENTED BY:
GROUP A
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THIS PRESENTATION
IS ALL ABOUT
“SETS”
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SETIn mathematics, a set is a collection of distinct objects, considered as an object
in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written
{2,4,6}. Sets are one of the most fundamental concepts in mathematics.
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EXAMPLES: 1:Set of names starting with letter A.
2:Set of members of math club.
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THERE ARE TWO DIFFERENT WAYS OR TYPES OF
WRIING SETS
1.Listing the elements2. Describing the set
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Listing the elementsWhen the set is given and we will write the elements inside the bracket.
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Describing the setWhen we describe a set in the form of a sentence
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1.Listing the elementsA=(1,2 ,3,4,5)
2.Describing the set Set of first five natural numbers or (x/x is a natural number less than six)
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SUBSETIf every member of set A is also a
member of set B, then A is said to be a subset of B, written A ⊆ B (also pronounced A is contained in B). Equivalently, we can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A.
A is a subset of B
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PROPER SUBSET
If A is a subset of, but not equal to, B, then A is called
a proper subset of B, written A ⊊ B (A is a proper subset of B) or B ⊋ A (B is a proper superset of A).
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*Note that the expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to mean the same as A ⊆ B (respectively B ⊇ A), whereas other use them to mean the same as A ⊊ B (respectively B ⊋ A).
Example:The set of all men is a proper subset of
the set of all people.{1, 3} ⊊ {1, 2, 3, 4}.{1, 2, 3, 4} ⊆ {1, 2, 3, 4}.
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The empty set is a subset of every set and every set is a subset of itself:
∅ ⊆ A.A ⊆ A.An obvious but useful identity, which
can often be used to show that two seemingly different sets are equal:
A = B if and only if A ⊆ B and B ⊆ A.
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BASIC OPERATIONS OF SET
1.COMPLEMENTARY 2.UNION 3. INTERSECTION
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COMPLEMENTARYTwo sets can also be "subtracted". The relative
complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B (or A − B), is the set of all elements which are members of A but not members of B. Note that it is valid to "subtract" members of a set that
are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect. In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply
complement of A, and is denoted by A′.
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Examples:{1, 2} \ {red, white} = {1, 2}.
{1, 2, green} \ {red, white, green} = {1, 2}.
{1, 2} \ {1, 2} = ∅.Y
A relative complement of B in A The complement of A in U
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UNIONTwo sets can be "added" together. The
union of A and B, denoted by A ∪ B, is the set of all things which are members of either A or B.
Examples:{1, 2} ∪ {red, white} ={1, 2, red, white}.{1, 2, green} ∪ {red, white, green} ={1,
2, red, white, green}.{1, 2} ∪ {1, 2} = {1, 2}. The union
of A and B
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INTERSECTIONA new set can also be constructed by
determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things which are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint.
The intersection of A and B, denoted A ∩ B.Examples:{1, 2} ∩ {red, white} = ∅.{1, 2, green} ∩ {red, white, green} = {green}.{1, 2} ∩ {1, 2} = {1, 2}. The
intersection of A and B
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• What is the definition of set?
• What is a describing set?
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