Question

Consider the statements:

$(1)$ $p$ and $q$ are both true
$(2)$ $p$ is true and $q$ is false
$(3)$ $p$ is false and $q$ is true
$(4)$ $p$ is false and $q$ is false.

How many of these imply the negation of the statement “ $p$ and $q$ are both true”?

$0$

$1$

$2$

$3$

$4$

Answer: D

Concepts:logical deduction

Difficulty rating: 1280

Small Hint:

The negation fails only when both statements are true

Big Hint:

Check which of the four listed truth assignments are not case $(1)$

Solution:

The negation of “ $p$ and $q$ are both true” holds whenever at least one of $p$ and $q$ is false. This occurs in cases $(2),$ $(3),$ and $(4),$ for a total of $3.$

Thus, the correct answer is D.

1962 AMC 12 Problem 30

Answer: D

Difficulty rating: 1280

Small Hint:

Big Hint:

Solution:

Problem 30 in Other Years