# Does $\leq$ imply $<$?

If I know for two numbers $$a$$ and $$b$$ that

$${a < b }$$

Then is it correct to say that

$$a \leq b$$

I know that the second statement is true as long as the first one is.

However is it true that

$$a \leq b \implies a

That is does less than or equal imply less than?

• The header question only matches one of the question in the body of your post. It is true that $a<b\implies a≤b$ . It is not true that $a≤b\implies a<b$. – lulu Feb 8 at 23:55
• @lulu The header question is precisely the question in the end of the post? – mrtaurho Feb 8 at 23:56
• @mrtaurho True. I guess both directions are asked in the body of the post. I will edit my first comment accordingly. – lulu Feb 8 at 23:57

## 5 Answers

You're correct in the first case.

But now consider the proposition $$a\leq b.$$ In this case if $$a=b=4$$, $$a\leq b$$ is true.

However given $$a< b$$, it is false that $$4<4$$. So it is false that $$a\leq b \to a\lt b$$ is always true.

$$1\leq 1$$ (since $$1=1$$) but does not hold that $$1<1$$.

No. Take $$a=b$$ to see that this does not hold in general. Clearly, $$a\le a$$ but surely not $$a.

No.

Take any number $$a$$ in your domain. Then $$a=a$$ (and so $$a\le a$$) but $$a\not.

No. $$a\le b$$ means "a less than or equal to b." It could be equal to, hence not $$a.