Question

Which of the following are equivalent to (!(x<15 && y>=3))?

a) (x>15 && y<=3)
b) (x>=15 && y < 3)
c) (x>=15 || y < 3)
d) (x>15 || y < 3)

Answer 1

The correct answer is (x>15 || y < 3)

Step-by-step explanation:

The original expression, (!(x<15 && y>=3)), can be rewritten as (x>=15 || y < 3) using DeMorgan's Laws.

Option (a), (x>15 && y<=3), is not equivalent because it requires both x to be greater than 15 and y to be less than or equal to 3. The original expression only requires either x to be greater than 15 or y to be less than 3.

Option (b), (x>=15 && y < 3), is also not equivalent because it requires both x to be greater than or equal to 15 and y to be less than 3. The original expression only requires either x to be greater than 15 or y to be less than 3.

Option (c), (x>=15 || y < 3), is the correct answer because it captures the logical equivalence of the original expression. It states that either x is greater than or equal to 15 or y is less than 3.

Option (d), (x>15 || y < 3), is not equivalent because it requires either x to be greater than 15 or y to be less than 3, but the original expression only requires one of those conditions to be true. The original expression does not allow for both conditions to be true simultaneously.

Therefore, the correct answer is (x>15 || y < 3).

Answer 2

The expression equivalent to !(x<15 && y>=3) is c) (x>=15 || y<3), which is found using De Morgan's laws to negate the original compound inequality.

Step-by-step explanation:

The question asks which expression is equivalent to !(x<15 && y>=3), which is a logical NOT operation applied to the condition (x<15 && y>=3). To find the equivalent expression, we use De Morgan's laws, which state that the negation of a conjunction is the disjunction of the negations. Therefore, !(x<15 && y>=3) is equivalent to (x>=15 || y<3).

The correct answer from the provided options is c) (x>=15 || y<3). This is because the NOT operation inverses both the relation and the logical operator. The '<' becomes '>=', '&&' (AND) becomes '||' (OR), and '>=' becomes '<'.