Tell whether each of the following is true for all sets A, B, or C. If false, give a counterexample.?
a. A-B=A- ∅
b. (A �¾B) ̅=(A ) ̅�¾B ̅
c. A �¿(B�¾C)=(A�¿B)�¾C
d. (A-B)�¿A=A
e. A-(B�¿C)=(A-B)�¿(A-C)
To determine whether each of the statements is true or false for all sets A, B, or C, we need to analyze the statements and provide counterexamples if they are false.
a. A - B = A - ∅:
This statement is true for all sets A and B. When we subtract an empty set (∅) from any set A, we are left with the original set A because there is nothing to remove. Therefore, A - B = A - ∅ is always true.
b. (A �¾B) ̅ = (A ) ̅ �¾ B ̅:
To evaluate this statement, we need to use the concept of complements. The complement of a set A, denoted by A ̅, contains all the elements not in A. If we consider the case where A = {1, 2} and B = {2, 3}, we can see that (A �¾B) ̅ = {1} ̅ = ∅, (A ) ̅ �¾ B ̅ = {1, 2} ̅ �¾ {3} ̅ = ∅ �¾ ∅ = ∅. Clearly, ∅ is not equal to ∅, so the statement (A �¾B) ̅ = (A ) ̅ �¾ B ̅ is false.
c. A �¿ (B �¾ C) = (A �¿ B) �¾ C:
To evaluate this statement, we need to understand the concept of set difference and intersection. The set difference A �¿ B contains all the elements in A that are not in B.
For example, let A = {1, 2}, B = {2, 3}, and C = {3, 4}.
A �¿ (B �¾ C) = A �¿ (B ∪ C) = {1}
(A �¿ B) �¾ C = {1} �¾ C = {1, 4}
Since {1} is not equal to {1, 4}, the statement A �¿ (B �¾ C) = (A �¿ B) �¾ C is false.
d. (A - B) �¿ A = A:
In this statement, we need to consider the set difference and intersection. The set (A - B) �¿ A contains all the elements in A - B that are also in A.
For example, let A = {1, 2, 3} and B = {2, 3}.
(A - B) �¿ A = {1} �¿ {1, 2, 3} = {1}
A = {1, 2, 3}
Since {1} is equal to {1}, the statement (A - B) �¿ A = A is true.
e. A - (B �¿ C) = (A - B) �¿ (A - C):
To evaluate this statement, we need to consider the set difference and intersection. The set A - (B �¿ C) contains all the elements in A that are not in the set (B �¿ C).
For example, let A = {1, 2, 3}, B = {2, 3}, and C = {3, 4}.
A - (B �¿ C) = {1, 2, 3} - {1, 4} = {2, 3}
(A - B) �¿ (A - C) = ({1} �¿ {1, 2, 3}) = {1, 2, 3} �¿ {1} = {2, 3}
Since {2, 3} is equal to {2, 3}, the statement A - (B �¿ C) = (A - B) �¿ (A - C) is true.
In summary:
a. True for all sets A and B.
b. False. Counterexample: A = {1, 2}, B = {2, 3}.
c. False. Counterexample: A = {1, 2}, B = {2, 3}, C = {3, 4}.
d. True for all sets A and B.
e. True for all sets A, B, and C.