Logical reasoning validity question

Question

Use truth tables to show whether the following arguments are valid. If the argument is valid, explain why. If the argument is not valid, give a counterexample.

1. A ∨ B

B

2. C ∨ D

D ∨ ~C

D

Answer 1

1. 
   

A | B | A V B

T | T | T

T | F | T

F | T | T

F | F | F

A ∨ B

B

is not valid because the 2nd row in the above truth table is a counterexample.

This is the case where A V B is true but B is False.

2.

C| D |~C | C V D | D V ~C | (C V D) ^ (D V `C)

T | T | F | T | T | T

T | F | F | T | F | F

F | T | T | T | T | T

F | F | T | F | T | F

C V D

D V ~C

D

is a valid argument based on the above truth table, where D = (C V D) ^ (D V ~ C) for all rows.