Question:

In $Z_7 - \{0\}$ under multiplication mod $7$, if $2^{-1} y \,3^{-1} = 5^{-1}$, then $y =$

Updated On: Jun 21, 2022
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  • 4
  • 6
  • none of these
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The Correct Option is B

Solution and Explanation

We know that $Z_7 = \{1, 2, 3, 4, 5, 6\}$
Here identity element is 1 and inverse of each element is given by,
$1^{-1} = 1, 2^{-1} = 4, 3^{-1} = 5, 4^{-1} = 2, 5^{-1} = 3, 6^{-1} = 6$
$\therefore \, 2^{-1} y\, 3^{-1} = 5^{-1} $ (given) multiplying by 2 and 3 on left and right respectively
$2\, 2^{-1} \, y\, 3^{-1} 3 = 2\, 5^{-1} 3$
$\Rightarrow \, y=[2?3?3]_{mod \, 7} \Rightarrow \, y=4$
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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.

  1. Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
  2. Roster form - {(1, 1), (2, 4), (3, 9)}
  3. Arrow Representation