Tell whether each of the following is true for all sets
A, B If false, give a counterexample.
a. A-B= A-0 (with slashed O)
b. AUB (with line over AUB) = AUB (with line over A and a line over B)
a. true
b. false
U = {1,2,3,4,5}
A = {1,2,3}
B = {2,4,5}
A' = {4,5}
B' = {1,3}
(AUB)' = {}
A'UB' = {1,3,4,5}
However A' ∩ B' = {}
To determine whether these statements are true for all sets A and B, we can use basic set theory concepts and logic. Let's evaluate each statement one by one:
a. A - B = A - Ø:
To check if this statement is true for all sets A and B, we need to verify if the difference between A and B is equal to the difference between A and the empty set (Ø).
A - B represents the set of elements that are in A but not in B, whereas A - Ø represents all the elements in A.
To find a counterexample, we need to find a case where A - B and A - Ø are not equal. Let's consider the following sets:
A = {1, 2, 3}
B = {1, 3}
Using these sets, we have:
A - B = {2} (Elements in A that are not in B)
A - Ø = {1, 2, 3} (All elements in A)
Since A - B ≠ A - Ø for this counterexample, we conclude that the statement is false for all sets A and B, as it does not hold in this case.
b. AUB̅ = A̅UB̅:
Here, AUB̅ represents the union of A and the complement of B, while A̅UB̅ corresponds to the complement of A union the complement of B.
To determine if this statement is true for all sets A and B, we must investigate if both expressions are equivalent in all cases.
To find a counterexample, we need to find a case where AUB̅ and A̅UB̅ are not equal. Let's consider the following sets:
A = {1, 2, 3}
B = {2, 3}
Using these sets, we have:
AUB̅ = {1, 2, 3} ∪ {1} = {1, 2, 3}
A̅UB̅ = {∅} ∪ {∅} = {∅}
Since AUB̅ ≠ A̅UB̅ for this counterexample, we conclude that the statement is false for all sets A and B, as it does not hold in this case.
In summary:
a. The statement "A - B = A - Ø" is false, as shown by the counterexample A = {1, 2, 3} and B = {1, 3}.
b. The statement "AUB̅ = A̅UB̅" is false, as shown by the counterexample A = {1, 2, 3} and B = {2, 3}.