Mathematical Re...

$$( p \wedge q ) \vee ( \sim p \wedge q ) \vee ( \sim q \wedge r ) =? $$

The component statements are:

p: You are wet when it rains.

q: You are wet when you are in river.

The compound statement of these component statements using appropriate connective is:

$$∼(p⇒q)⟺∼p\vee ∼q  \, is$$

Name the technique used in the solution of the problems below :

Question: Show that the following statement is false: If n is an odd integer, then n is prime.

Solution: The given statement is in the form “if p then q” we have to show that this is false, If p then ~q.

If p and q are mathematical statements, then in order to show that the statement p and q is true, we need to show that:

$$[(p)\wedge q]$$ is logically equivalent to

Name the technique used in the first step of the solution to the problem below :
Verify that 5 is irrational
Solution : Let us assume that 5 is rational

An electrical circuit for a set of 4 lights depends on a system of switches A, B, C and D. When these switches work they have the following effect on the lights: They each toggle the state of two lights (i.e. on becomes off and off becomes on). The lights that each switch controls are as follows.

A	B	C	D
1 and 2	2 and 4	1 and 3	3 and 4

In configuration I shown below, switches C-B-D-A are asserted in order, resulting in configuration 2. One switch did not work and had no effect at all. Which was that switch?

The converse of $$\sim p \rightarrow q$$ is equivalent to

Let P(n) denote the statement that $$n^2+n$$ is odd. It is seen that $$P(n)\Rightarrow P(n+1), P(n)$$ is true for all.