Which of the following is a subgroup of the group $G = \{1, 2, 3, 4, 5, 6\}$ under $\otimes_7$ ?
- $\{2, 6, 1\}$
- $\{1, 2, 4\}$
- $\{5, 4, 2\}$
- $\{2, 3 , 1\}$
The Correct Option is B
Solution and Explanation
(a) $\{2, 6, 1) $ is not a subgroup
$\because 2\oplus_7 6=5 \notin \{ 2,6, 1 \}$
(b) $\{1, 2, 4\}$ is a subgroup
$\because$ it satisfies all properties of group.
$ 1\oplus_7 2=2, 2 \oplus_7 4=1,4 \oplus_7 1=4$
(c) $ \{5, 4, 2 \}$ is not a subgroup ?
$\because $ In this set identity element do not exist.
(d) $ \{2, 3, 1\} $ is not a subgroup
$\because 2 \oplus_7 3 = 6 \notin \{2, 3, 1\}$
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Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
