Using the laws of logic to prove tautologies. use the laws of propositional logic to prove that each statement is a tautology. (a) (p ∧ q) → (p ∨ r) (b) p → (r → p)

See explanation below.

Explanation:

If the statement is a tautology is true for all the possible combinations and we can check this with the table of truth for the statements

Part a

lets call this condition (1)

condition (2) and  condition (3)

We can create a table like this one:

p       q     r      (2)       (3)     (1)  

T       T     T      T        T       T

T       T     F      T        T       T        

T       F     T      F        T       T

T       F     F      F        T       T

F       T     T      F        T       T

F       T     F      F        F       T

F       F     T      F        T       T

F       F     F      F        F       T

So as we can see we have a tautology since for all the possibilites we got true the final result.

Part b

let's call this condition (1)

And let condition (2)

We can create the following table:

p     r       (2)     (1)

T     T       T       T

T     F       T       T

F     T       F       T

F     F       T       T

So is also a tautology since the statement is true for all the possibilities or combinations.

solution:

as, mean is defined as sum of all the observations divided by total number of observations.

and , median is mid value of the data whether arranged in ascending order or descending order.

consider, roadsters:

2,3,4,5,6,7,8,9

mean =

also, median= , because number of observations is even, so median is mean of two mid values.

now, coming to bandits

2,3,4,5,6

mean =  

median = 4, as number of observations is odd.

so, roadsters has greater median as well mean than bandits.

→option a is true about the repairs performed on 2 types of cars:

bandits have lower median and mean than roadsters.

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