In Maths, Set theory was developed to explain about collections of objects. Basically, the definition states it is a collection of elements. These elements could be numbers, alphabets, variables, etc. The notation and symbols for sets are based on the operations performed on them. You must have also heard of subset and superset, which are the counterpart of each other. The different types of sets in Mathematics set theory are explained widely with the help of Venn diagrams.
Sets have turned out to be an invaluable tool for defining some of the most complicated mathematical structures. TheyÂ are mostly used to define many real-life applications. Apart from this, there are also many types of sets, such as empty set, finite and infinite set, etc.
What is Set Theory in Maths?
As we have already discussed, in mathematics set theory, a set is a collection for different types of objects and collectively itself is called an object. For example, number 8, 10, 15, 24 are 4 distinct numbers, but when we put them together, they form a set of 4 elements, such that, {8, 10, 15, 24}.
In the same way, sets are defined in the Maths for a different pattern of numbers or elements. Such as, sets could be a collection of odd numbers, even numbers, natural numbers, whole numbers, real or complex numbers and all the set of numbers which comes in the number line.
Mathematics Set Theory Symbols
Let us see the different types of symbols used in Mathematics set theory with its meaning and examples.Â Consider a Universal set (U) = {1, 2, 7, 9, 13, 15, 21, 23, 28, 30}
Symbol |
Symbol Name |
MeaningÂ |
Example |
{ } | set | a collection of elements | A = {1, 7, 9, 13, 15, 23},
B = {7, 13, 15, 21} |
A âˆª B | union | Elements that belong to set A or set B | A âˆª B = {1, 7, 9, 13, 15, 21, 23} |
A âˆ© B | intersection | Elements that belong to both the sets, A and B | A âˆ© B = {7, 13, 15 } |
A âŠ† B | subset | subset has few or all elements equal to the set | {7, 15} âŠ† {7, 13, 15, 21} |
A âŠ„ B | not subset | left set not a subset of right set | {1, 23} âŠ„ B |
A âŠ‚ B | proper subset / strict subset | subset has fewer elements than the set | {7, 13, 15} âŠ‚ {1, 7, 9, 13, 15, 23} |
A âŠƒ B | proper superset / strict superset | set A has more elements than set B | {1, 7, 9, 13, 15, 23} âŠƒ {7, 13, 15, } |
A âŠ‡ B | superset | set A has more elements or equal to the set B | {1, 7, 9, 13, 15, 23} âŠƒ {7, 13, 15, 21} |
Ã˜ | empty set | Ã˜ = { } | C = {Ã˜} |
P (C) | power set | all subsets of C | C = {4,7},
P(C) = {{}, {4}, {7}, {4,7}} Given by 2^{s}, s is number of elements in set C |
A âŠ… B | not superset | set A is not a superset of set B | {1, 7, 9, 13, 15, 23} âŠ…{7, 13, 15, 21} |
A = B | equality | both sets have the same members | {7, 13,15} = {7, 13, 15} |
A \ B or A-B | relative complement | objects that belong to A and not to B | {1, 9, 23} |
A^{c} | complement | all the objects that do not belong to set A | We know, U = {1, 2, 7, 9, 13, 15, 21, 23, 28, 30}
A^{c} = {2, 21, 28, 30} |
A âˆ† B | symmetric difference | objects that belong to A or B but not to their intersection | A âˆ† B = {1, 9, 21, 23} |
aâˆˆB | element of | set membership | B = {7, 13, 15, 21},
13 âˆˆ B |
(a,b) | ordered pair | collection of 2 elements | |
xâˆ‰A | not element of | no set membership | A = {1, 7, 8, 13, 15, 23}, 5 âˆ‰ A |
|B|, #B | cardinality | the number of elements of set B | B = {7, 13, 15, 21}, |B|=4 |
AÃ—B | cartesian product | set of all ordered pairs from A and B | {3,5} Ã— {7,8} = {(3,7), (3,8), (5,7), (5, 8) } |
N_{1} | natural numbers / whole numbers Â set (without zero) | N_{1} = {1,2,3,4,5,â€¦} | 6 âˆˆ N_{1} |
N_{0} | natural numbers / whole numbers Â set (with zero) | N_{0} = {0,1,2,3,4,â€¦} | 0 âˆˆ N_{0} |
Q | rational numbers set | Q= {x | x=a/b, a,bâˆˆZ} | 2/6 âˆˆ Q |
Z | integer numbers set | Z= {â€¦-3,-2,-1,0,1,2,3,â€¦} | -6 âˆˆ Z |
C | complex numbers set | C= {z | z=a+bi, -âˆž<a<âˆž, Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â -âˆž<b<âˆž} | 6+2iÂ âˆˆ C |
R | real numbers set | R= {x | -âˆž < x <âˆž} | 6.343434 âˆˆ R |
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