logic

[Ms. L]

If a then b and if not c then not b are both given statements, which statement must be true:

if a then c

if b then a

if c then a

if c then b

 

[Deleted member 4993]

If a then b and if not c then not b are both given statements, which statement must be true:

if a then c

if b then a

if c then a

if c then b

First, write contra-positive of the highlighted statement.

Then combine the other given statement with it (contra-positive).

 

[soroban]

Hello, Ms. L!

It would help if I knew what laws of Logic you know.

\(\displaystyle \text{If }\,a \to b\,\text{ and }\sim\!c \to\,\sim\!b\) .\(\displaystyle \text{ are given statements, which statement must be true?}\)

[COLOR=#e00e]. . [/COLOR]\(\displaystyle (1)\;a \to c \qquad (2)\;b \to a \qquad (c)\;c \to a \qquad (d)\;c \to b\)

We have: .\(\displaystyle \begin{array}{c}a\to b \\ \sim\!c \to\,\sim\!b \end{array}\)


\(\displaystyle \text{Since }\sim\!c\to\, \sim\!b \;\equiv\;b\to c\)

\(\displaystyle \text{We have a syllogism }\:\begin{array}{cc}& a \to b \\ & b \to c \\ \hline \therefore & a \to c \end{array}\;\;\text{ answer (1)}\)

 

[Ms. L]

Thank you!

 

[lookagain]

If a then b and if not c then not b are both given statements,
which statement must be true:

if a then c

if b then a

if c then a

if c then b

Here are examples of a, b, and c that can help illustrate the situation:


Let a = the number is an integer

Let b = the number is a rational number

Let c = the number is a real number