Question

One of the following is false

(A) A∪B=x:x∈Aorx∈B
(B) A\(B U C) = (A\B)n(A\C)
(C) A\(B U C)= (A\B)U(A\C)
(D) A\(B n C) = (A\B)U(A\C)

Answer 1

Conditions

One of the following is false

(A) A \UB=x:x\in A or x\in B

(B) A\B \U C = (A\B)n(A\C)

(C) A\B \U C = (A\B)U(A\C)

(D) A\B n C = (A\B)U(A\C)

Please explain

Solution

The false statement is C.

Explanation:

Consider point b from set A and B but not from C:

$$b \in A \cap B, b \cap C = \emptyset$$

Consider set $A \setminus (B \cup C)$. $b$ is not from this set, as $b \in A \cap B, \rightarrow, b \in B$:

$$b \cap A \setminus (B \cup C) = \emptyset$$

Consider set $(A\backslash B)\cup (A\backslash C)$. The first set doesn't have point b in it, as b is from B, but the second set has this point in, because b is from A, but b isn't from C, so if we exclude all C points from A, there will remain our point b. As we have a $\cup$ between, it means, that there is point b in $(A\backslash B)\cup (A\backslash C)$

So, on the left side of this set equation we have a set, which doesn't include point b and on the right side – a set which includes this point. **Here is a false.**