Fisher Test in R Suppose we have the following data set: 
                Men    Women    
Dieting         10      30
Non-dieting     5       60

If I run the Fisher exact test in R then what does alternative = greater (or less) imply? For example: 
mat = matrix(c(10,5,30,60), 2,2)
fisher.test(mat, alternative="greater")

I get the p-value = 0.01588 and odds ratio =  3.943534. Also, when I flip the rows of the contingency table like this:
mat = matrix(c(5,10,60,30), 2, 2)
fisher.test(mat, alternative="greater")

then I get the p-value = 0.9967 and odds ratio = 0.2535796. But, when I run the two contingency table without the alternative argument (i.e., fisher.test(mat)) then I get the p-value = 0.02063. 


*

*Could you please explain the reason to me? 

*Also, what is the null hypothesis and alternative hypothesis in the above cases?

*Can I run the fisher test on a contingency table like this: 
mat = matrix(c(5000,10000,69999,39999), 2, 2)

PS: I am not a statistician. I am trying to learn statistics so your help (answers in simple English) would be highly appreciated. 
 A: greater (or less) refers to a one-sided test comparing a null hypothesis
that p1=p2 to the alternative p1>p2 (or p1<p2). In contrast, a two-sided
test compares the null hypotheses to the alternative that p1 is not equal to
p2.
For your table the proportion of dieters that are male is 1/4 = 0.25 (10 out
of 40) in your sample. On the other hand, the proportion of non-dieters that are
male is 1/13 or (5 out of 65) equal to 0.077 in the sample.  So then the
estimate for p1 is 0.25 and for p2 is 0.077. Therefore it appears that
p1>p2.
That is why for the one-sided alternative p1>p2 the p-value is 0.01588.
(Small p-values indicate the null hypothesis is unlikely and the alternative is
likely.)
When the alternative is p1<p2 we see that your data indicated that the
difference is in the wrong (or unanticipated) direction.
That is why in that case the p-value is so high 0.9967.  For the two-sided
alternative the p-value should be a little higher than for the one-sided
alternative p1>p2. And indeed, it is with p-value equal to 0.02063.
