# Which statement form is right?

In The book "Discrete mathematics and its application" - by Susanna Epp, it said

$$x\ge a$$ means $$x \gt a$$ or $$x = a$$

so, in one of the question we where given.

write this in statement form:-

$$x \ge 3$$

$$sol^n$$:-

let p be $$x \gt 3$$ and q be $$x = 3$$

here is my answer, $$p \ XOR \ q$$

and in the book's solution it wrote, $$p \lor q$$

I am sure that it should be the exclusive "or" or maybe i am wrong ?

• $x\ge a ~$ means $~x \gt a ~~\color{red}{\text{or}} ~~x = a.$ Commented Jul 17 at 8:08
• so like, are both answers correct ? is that what u mean ? Commented Jul 17 at 8:10
• I am not allowed to say more, because your posted question is low quality with respect to MathSE protocol. For one thing, what exactly does the expression $~p \oplus q~$ signify? For another, see next comment. Commented Jul 17 at 8:13
• For what it's worth, virtually every MathSE posted question that I have seen, that followed this article on MathSE protocol has been upvoted rather than downvoted. I am not necessarily advocating this protocol. Instead, I am merely stating a fact: if you scrupulously follow the linked article, skipping/omitting nothing, you virtually guarantee a positive response. Commented Jul 17 at 8:14
• The textbook - up to that page - has introduced the "usual" propositional connectives: $\land, \lor$ etc. and thus we imagine to use them. Commented Jul 17 at 8:15

The only difference between "$$P$$ or $$Q$$" and "$$P$$ xor $$Q$$" is that if $$P$$ and $$Q$$ are both true, then "$$P$$ or $$Q$$" is true but "$$P$$ xor $$Q$$" is false. That means that for the statement in your question, the only difference between "$$x > a$$ or $$x = a$$" and "$$x > a$$ xor $$x = a$$" is that if $$x > a$$ and $$x = a$$ are both true, then the first statement is true and the second is false. But of course that situation is impossible--$$x > a$$ and $$x = a$$ cannot both be true. So it doesn't matter which way you write it; the truth value will come out right either way. Most mathematicians use "or" in this situation.
To put it another way: "$$P$$ xor $$Q$$" means "either $$P$$ or $$Q$$ but not both". Most mathematicians don't add the extra restriction "but not both" unless it is needed. In this case, it's not needed, because $$x>a$$ and $$x=a$$ both being true is impossible, so there's no need to explicitly rule it out.
$$p\oplus q$$ this would imply that $$x>3$$ or $$x=3$$ , but not both simultaneously. While in practical terms, $$x$$ cannot be both greater than and equal to $$3$$ at the same time, the exclusive OR operator introduces a level of restriction that isn't necessary for this problem.
• @Anonymous The restriction is the narrowing that occurs when changing from the first connector $~\color{red}{\text{or}}~$ to the second connector $~\color{red}{\text{exclusive or}}.~$ The second connector is the exact same as the first connector, except that the second connector does not allow both the LHS and the RHS to both be (simultaneously) true. Commented Jul 17 at 8:16
• @Anonymous XOR is a stricter condition than OR. $x$being greater than 3 or equal to 3 are typically considered as mutually exclusive (they cannot happen at the same time). Therefore, the XOR operator introduces a stricter logical condition than what is naturally implied by the statement "$x>3 \text{or} x=3$" alone Commented Jul 17 at 8:17