# Math Help - Truth Tables and Euler Circles

1. ## Truth Tables and Euler Circles

I am having a hard time understanding truth tables. Anyone able to show me a truth table for the following 2 problems?

a.) no A is B
some C is A
------------
therefore some C is not B

B.) All B is A
All C is A
-----------
therefore All C is B

help with these will be greatly appreciated I am having major issues understanding truth tables and my work is overdue.

There seem to be two major difficulties:
. . [1] Translating statements into logic
. . [2] Constructing and completing truth tables.
Can you describe yours?

1) .No A is B . . . . . . . ${\color{blue}a \to \sim b}$
. . .Some C is A . . . . . ${\color{blue}c \wedge a}$
. . .------------------- . ---------
. $\therefore$ Some C is not B . . ${\color{blue} c \:\wedge \sim b}$

$\begin{array}{|c|c|c||ccccccccccc|}
a & b & c & [(a & \to & \sim b) & \wedge & (c & \wedge & a)] & \to & (c & \wedge & \sim b) \\ \hline \hline
T&T&T & T&F&F &F& T&T&T & T & T&F&F \\
T&T&F & T&F&F &F& F&F&T & T & F&F&F \\
T&F&T & T&T&T &T& T&T&T &T& T&T&T \\
T&F&F & T&T&T &F& F&F&T &T& F&F&T \\
F&T&T & F&T&F &F& T&F&F &T& T&F&F \\
F&T&F& F&T&F &F& F&F&F &T& F&F&F \\
F&F&T & F&T&T &F& T&F&F &T& T&T&T \end{array}$

$\begin{array}{|c|c|c||ccccccccccc|}F&F&F & F&\;T&\;\;\;T &\;\;F& F&F&F &\;\;T& F&\;T&\;\;T\;\; \\ \hline \hline
& & & 1 & 2 & 1 & 3 & 1 & 2 & 1 & 4 & 1 & 2 & 1 \end{array}$

The last column (4) is all T's.
The statement is valid.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Now try the second problem . . .

2) All B is A . . ${\color{blue} b \to a}$
. . All C is A . . ${\color{blue}c \to a}$
. . ----------- . .-------
. $\therefore$ All C is B . . ${\color{blue}c \to b}$

Now test: . $\bigg[(b \to a) \wedge (c \to a)\bigg] \to (c \to b)$