Question

Specify the set of all integers that are congruent to 5 (mod 7) using (a) the roster method and (b) set-builder notation

Answer 1

(a) A = {... -23, -16, -9, -2, 5, 12, 19, 26, ....}

(b)
`A = {x\in Z: x = 5 - 7t,\ t\in Z}`

Solution:

For two integers x and y to be congruent, we know that:

If 't' divides y - x, we say that x is congruent to y modulo t, written x ≡ y mod t.

Now,

In roster form, we include each element in the representation of the set.

In set-builder form, we use mathematical notation and properties of the elements of the set.

Now,

As per the question;

(a) To represent all the integers congruent to 5 (mod 7) by roaster method of a set A (say):

where

`x\in Z` such that

x ≡ 5 (mod 7)

Then

A = {... -23, -16, -9, -2, 5, 12, 19, 26, ....}

(b) In set builder form,

`A = {x\in Z: x = 5 - 7t,\ t\in Z}`