Tell whether each of the following is true for all sets
A, B, or C. If false, give a counterexample.
c. A�¿(BUC) = (A�¿B) UC
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 can use set theory concepts and logic.
c. A�¿(BUC) = (A�¿B) UC (proposed statement)
To test the statement, we need to find a counterexample – a scenario where the statement doesn't hold true. Let's consider the following example:
A = {1, 2}
B = {2, 3}
C = {3, 4}
Now let's evaluate both sides of the equation for this example:
Left-hand side: A�¿(BUC)
{(1, 2)} �¿ {(2, 3, 4)} = {} (empty set)
Right-hand side: (A�¿B) UC
({1}�¿{2, 3}) UC = {1} UC = {1} (set containing 1)
Since the left-hand side (empty set) is not equal to the right-hand side (set containing 1), the statement is false for this counterexample. Hence, the statement "A�¿(BUC) = (A�¿B) UC" is false for all sets A, B, or C.
d. (A-B)�¿A=A (proposed statement)
To test this statement, let's consider a counterexample:
A = {1, 2, 3}
B = {2, 3}
Let's evaluate both sides of the equation for this example:
Left-hand side: (A-B)�¿A
{1} �¿ {1, 2, 3} = {} (empty set)
Right-hand side: A
{1, 2, 3} (set containing 1, 2, and 3)
Since the left-hand side (empty set) is not equal to the right-hand side (set containing 1, 2, and 3), the statement is false for this counterexample. Therefore, the statement "(A-B)�¿A=A" is false for all sets A, B, or C.
e. A-(B�¿C) = (A-B) �¿ (A-C) (proposed statement)
To test this statement, let's consider a counterexample:
A = {1, 2}
B = {1}
C = {2}
Let's evaluate both sides of the equation for this example:
Left-hand side: A-(B�¿C)
{1, 2}-(1�¿2) = {1} (set containing 1)
Right-hand side: (A-B) �¿ (A-C)
({1, 2}-{1}) �¿ ({1, 2}-{2}) = {2} �¿ {1} = {} (empty set)
Since the left-hand side (set containing 1) is not equal to the right-hand side (empty set), the statement is false for this counterexample. Hence, the statement "A-(B�¿C) = (A-B) �¿ (A-C)" is false for all sets A, B, or C.