The truth table represents statements p, q, and r. p q r a t t t b t t f c t f t d t f f e f t t f f t f g f f t h f f f which statements are true for rows a and e? check all that apply. p ↔ q p ↔ r q ↔ p q ↔ r r ↔ p r ↔ q

We have been given the truth table for three variables p, q, and r. as shown below

p q r

A T T T

B T T F

C T F T

D T F F

E F T T

F F T F

G F F T

H F F F

Now we need to find Which statements are true for rows A and E for the following statements:

p ↔ q p ↔ r q ↔ p q ↔ r r ↔ p r ↔ q

To find that we need to use table of P <-> Q as shown in picture.

We know that if both statements are true for P and Q then only P <-> Q will be true. So using that trick we see that only q<->r and r<->q are the final answer as shown in the attached picture.

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!