Question

If AB is a subset of C, Show that P (C)≥P(A) +P(B) - 1. Please show me all of your work ​

Answer 1

Answer:

The relation R on set A={1,2,3,4,5,6,7} is defined by

R={(a,b): both a and b are either odd or even}

We observe the following properties of R on A

Reflexivity: Clearly, (1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7)∈R. So, R is a reflexive relation in A

Symmetric: Let a,b∈A be such that (a,b)∈R.

Then, (a,b)∈R

Both a and b are either odd or even

Both b and a are either odd or even

⇒ (b,a)∈R

Thus, (a,b)∈R⇒(b,a)∈R for all a,b∈A

So, R is a symmetric relation on A

Transitivity: Let a,b,c∈Z be such that (a,b)∈R,(b,c)∈R.

Then, (a,b)∈R⇒ Both a and b are either odd or even

(b,c)∈R⇒ Both b and c are either odd or even

If both a and b are even, then

(b,c)∈R⇒ Both b and c are even

If both a and b are odd, then

(b,c)∈R⇒ Both b and c are odd

∴ Both a and c are even or odd. Therefore (a,c)∈R

So, (a,b)∈R and (b,c)∈R⇒(a,c)