My dad's mom had breast cancer and now my mom had breast cancer. Even though I am not yet 40 should I have a mammogram?

High risk evaluation. I recommend you seek out a high risk breast program or specialist to do an extensive family history to determine when and what type of screening is indicated.
Maybe. You should ask your doctor to refer you to a genetic counselor to evaluate your long term risk for breast cancer. Depending on your risk, there may be a role for genetic testing for genetic predisposition for breast cancer in your or your mother. In some cases, it makes sense to do annual breast mris, as they are more sensitive than mammography in younger women.
Need more info. I'm sure that it feels that you have a "double dose" of risk, but, if these are your only family members with breast (or ovarian) ca, you are not high-risk for hereditary breast ca (we tend to see this type in families with multiple members with ca, especially prior to menopause or when in both breasts) & can wait till 40. That said, a mammogram is a small price to pay for peace of mind.
You should have. a comprehensive risk assessment, then you and your doctor can assess whether it would be appropriate to begin early screening.
More to my answer. Dear patient, mammogranms are very important. But, mammograms are not created equal. The type of mammo machine, the technician who does the mammogram, the radiologist who reads the mammogram, the physician who reads the report, the report being discussed with you in detail, and appropriate recommendations following the report based on your exam and history. All affect the reliabilty of the mammo.
YES. I really don't have enough famiy history, but baseline mammograms can be done between 35-40 years anyway-without extra family history.Please also note if you ever feel unusual senastions-unequally in your breast-not pain really but tingling in onebreast that persists for month - any nipple dischatge esp only in one breast. Go to a physician/ medical center known for tx/prev. Of breast disorders.
Yes. Generally, patients who have a first degree relative should start annual mammograms starting at an age 10 years younger than your mother was when she was diagnosed. I would recommend you start.