I would never assume that my children are invited to a wedding. I think of the 20 weddings I've attended in my life, only one or two were family friendly. The rest were all adults. I would probably assume I should bring a date to a wedding... but either way, you can clarify when you send the invites.
True story: I got married 12 years ago. When we sent out our invites, we hated the inner envelopes so we just stuck all of the various pieces of paper (directions, reply card, etc.) in the outer envelope with the actual invite. Most of the people we were inviting were either married or lived with their significant others, so it was easy to write both names on the outer envelope. I never really thought about those few people who were dating someone but not living together. I assumed they'd realize they could bring a date, even though I hadn't written "Jenny and guest" on the inner envelope -- since we didn't have one. All of my friends who fell into this category called to confirm that they could bring their girl/boyfriends. And of course they could! But I had one cousin who was seriously dating a woman that I had met a few times and really liked. He was living with his parents on the other side of the country, so I just addressed the invite to him, thinking he'd know that Allison was invited. But after all of my friends called, I asked my cousin's sister, who lived near me, to make sure he knew that Allison was invited. Well... she didn't. In fact, she told him that if I hadn't put Allison's name on the invite, she wasn't invited. So my male cousin didn't show up to my wedding in protest. And his parents were so offended that I had hurt their son and future daughter in law like this. Oh my gosh - it was so much drama. All because I didn't like the inner envelopes (well - and because my female cousin turned out to be such a jerk).
So please - don't make the mistake I did. Spell it out for people on the invites.