Mathematical Reasoning

Question

I can't understand the answers provided for these 2 question in a textbook:

Question 1 : State whether true or false

a) Implication : If 4 is a root of x² - 16 = 0, then 4 is not a root of ( x + 4 ) ( x - 4 ) = 0 (answer: false)

b) converse : If 4 is not a root of ( x + 4 ) ( x - 4 ) = 0, then 4 is a root of x² - 16 = 0 (answer: true) ( ? )

c) inverse : If 4 is not a root of x² - 16 = 0, then 4 is a root of ( x + 4 ) ( x - 4 ) = 0 (answer: true) ( ? )

d) contrapositive: If 4 is a root of ( x + 4 ) ( x - 4 ) = 0, then 4 is not a root of x² - 16 = 0 (answer: false)

Why are the answers provided for (b) and (c) are "true"? Shouldn't it be false ?

Question 2 : State whether true or false

a) Implication : If 55 + 55 = 4 x 5, then 666 + 666 = 6 x 6 (answer: true)

b) converse : If 666 + 666 = 6 x 6 , then 55 + 55 = 4 x 5 (answer: true)

c) inverse : If 55 + 55 ≠ 4 x 5, then 666 + 666 ≠ 6 x 6 (answer: true)

d) contrapositive: If 666 + 666 ≠ 6 x 6 , then 55 + 55 ≠ 4 x 5 (answer: true)

I don't understand the logic of question 2. Why are all the statements true ?

Answer 1

First, you need to understand the truth table for the IF-Then statement (a.k.a the implication)

Case #1: IF true then true ---> the entire statement is TRUE

Case #2: IF false then true ---> the entire statement is TRUE

Case #3: IF true then false --> the entire statement is FALSE

Case #4: IF false then false --> the entire statement is FALSE...

#1)

Now, 4 is a root of x^2-16=0 is a TRUE statement

4 is a root of (x+4)(x-4) = 0 is a TRUE statement

(a) is case #3 so the entire statement is FALSE

(b) is case #2 so the entire statement is TRUE

(c) is case #2 so the entire statement is TRUE

(d) is case #3, so the entire statement is FALSE

#2) 55+55 = 4 * 5 is false

666+666 = 6*6 is false

(a)-(b) are case #4 so the entire statement is true

(c) -(d) are case #1, so the entire statement is true