ব্যাংক জব নিয়োগ পরীক্ষা গণিত Set
Set: A set is a collection of objects together with some rule to determine whether a given object belongs to this collection. Any object of this collection is called an element of the set.
Notation: The name of a set is denoted with a capital letter – A, B, The description of the set can be given in the following ways:
1. Each element of the set is listed within a set of brackets: { }.
2. Within the brackets, the first few elements are listed, with dots following to show that the set continues with the selection of the elements following the same rule as the first few.
3. Within the brackets, the set is described by writing out the exact rule by which elements are chosen. The name given each element is separate from the selection rule with a vertical line.
Examples:
(a) Denote by A the set of natural numbers with are greater than 25. The set could be written in the following ways:
A={26,27,28….} (using the second notation listed above)
A={x | x is a natural number and x > 25} (using the third notation above)
The above description is read as “the set of all x such that x is a natural number and x > 25”.
Note that 32 is an element of A. We write 32 ∈ A, where “∈” denotes “is
an element of.” Also, 6 ∉ A, where “∉” denotes “is not an element of.”
(b) Let B be the set of numbers {3,5,15,19,31,32}. Again the elements of the set are natural numbers. However, the rule is given by actually listing each element of the set (as in the first notation above). We see that 15 ∈ B, but 23 ∉ B.
(c) Let C be the set of all natural numbers which are less than 1. In this set, we observe that there are no elements. Hence, C is said to be an empty set. A set with no elements is denoted by ∅.
Subset: A set A is said to be a subset of a set B if every element of A is an element of B.
Notation: To indicate that set A is a subset of set B, we use the expression A ⊂ B, where “⊂” denotes “is a subset of”. A ⊄ B means that A is not a subset of B.
Examples:
(a) Let B be the set of natural numbers. Let A be the set of even natural numbers. Clearly, A is a subset of B. However, B is not a subset of A, for 3 ∈ B, but 3 ∉ A.
(b) An empty set ∅ is a subset of any set B. If this were not so, there would be some element x ∈ ∅ such that x ∉ B. However, this would contradict with the definition of an empty set as a set with no elements.
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