Specify the set of all integers that are congruent to 5 (mod 7) using (a) the roster method and (b) set-builder notation
1 answer:
Answer:
(a) A = {... -23, -16, -9, -2, 5, 12, 19, 26, ....}
(b)
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
such that
x ≡ 5 (mod 7)
Then
A = {... -23, -16, -9, -2, 5, 12, 19, 26, ....}
(b) In set builder form,
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Answer:
(a) F(A, B, C) = AB'C' + A'BC' + A'B'C
(b) F(A, B, C) = (A' + B' + C')(A' + B + C)(A + B' + C)(A + B + C')(A + B + C)
Step-by-step explanation:
(a) If F(A, B, C) = 1 iff exactly one of the coins is heads, then either
A is heads and the others are tails (AB'C')
B is heads and the others are tails (A'BC')
C is heads and the others are tails (A'B'C)
Hence, as a minterm expansion,
F(A, B, C) = AB'C' + A'BC' + A'B'C
(b) To get the corresponding maxterm expansion, we convert to binary.
The maxterm is the product of the complements.
Expanding,
F(A, B, C) = (A' + B' + C')(A' + B + C)(A + B' + C)(A + B + C')(A + B + C)