Question

Complete the following statements.

1. A conjunction is true (only if both statements are true, if at least one statement is true)

2. A disjunction is true (only if both statements are true, if at least one statement is true)

3. For a conditional statement, a (false hypothesis cannot imply a true conclusion, true hypothesis cannot imply a false conclusion)

Answer 1

• A conjunction is true only if both statements are true.

The conjunction is the AND logical operator, and the result of A AND B is true only when both A and B are true. For example, "We are on planet earth, and elephants fly" is false, even if the first part is actually true.

• A disjunction is true if at least one statement is true

In fact, the disjunction is the OR operator, and this time one true statement is sufficient to make the whole expression true: "You are reading this answer on brainly or I am 4 meters tall" is true, although I'm not 4 meters tall.

• For a conditional statement, a true hypothesis cannot imply a false conclusion

In fact, the latin expression "ex falso sequitur quodlibet" translates as "you can deduce whatever you want from a false premise". Using formulas, this means that the implication $A \implies B$ is always true, no matter the truth value of B. So, both these implications

$1+2 = 4 \implies \pi = 8$

$1+2 = 4 \implies 10-3=7$

are true.

Conversely, if the hypothesis is true, then the implication is true only if the conclusion is true as well: if we change the previous example to

$1+2 = 3 \implies \pi = 8$

$1+2 = 3 \implies 10-3=7$

Only the second implication will hold.