The question is whether or not Jamie is an element of .

Assume both statements to be true. Then Jamie is an element of the set , shaded in this Venn diagram:

Some elements of  are elements of , but some are not, making the two statements together insufficient to answer the question of whether Jamie is an element of . Whether Jamie was born in an even month or not cannot be determined.

 is the intersection of , the complement of the set of people who like cheddar cheese - that is, the set of people who don't like cheddar cheese - and , the set of people who like Swiss cheese. For  to be true, it must hold that Gina doesn't like cheddar cheese and Gina likes Swiss cheese. Either statement alone makes this false.

Assume Statement 1 alone.  is the intersection of two sets -  and , the complements of, respectively, the set of people who like The Eagles and the set of people who like Bruce Springsteen. Equivalently, this is the intersection of the set of people who do not like The Eagles and the set of people who do not like Bruce Springsteen. Since , Holly falls in both sets. Specifically, she does not like The Eagles.

Assume Statement 2 alone.  is the union of two sets - , the set of people who do not like The Eagles and , the set of people who like Crosby, Stills, and Nash. If , Holly falls in either or both sets, so either she doesn't like The Eagles, or she likes Crosby, Stills, and Nash, or both. It is not resolved whether she likes The Eagles or not.

Assume both statements to be true.

 is the union of the sets , , and ; Veronica falls in this union, so she falls in at least one of the three sets, so she likes one, two, or all three of Pearl Jam, Nirvana, and Soundgarden.

Similarly, from Statement 2, Veronica falls in at least one of the three sets , , and  - the complements of the three sets - meaning that she doesn't like at least one of the three. Therefore, Veronica likes none, one, or two of the artists.

From the two statements together, it follows that Veronica likes one or two of the artists. Without further information, however, it cannot be determined whether she likes two of the artists.

 is the union of the set , the set of all people who like coffee, and , the complement of the set of all people who like tea - that is, the set of all people who do not like tea.  if either Frank likes coffee or Frank does not like tea or both. 

Statement 1 alone, that Frank likes tea, does not prove or disprove this to be true true; if Frank likes coffee, then this is true, and if he does not like coffee, then this is not true. Statement 2 alone, however, proves this statement true.

 and .

This can be respresented in a Venn diagram as follows:

Assume both statements to be true. If we let  represent Bumpy, then from the diagram, it can be seen that if Bumpy is neither a Bee or a Dee, it is possible for Bumpy to be be a Fee or to not be a Fee.

Let  be the sets of Fe's, Fi's, Fo's and Fum's, respectively. According to the premise of the problem, the sets  and  are disjoint (they have no elements in common); the sets  and  are disjoint; and the sets  and  are disjoint. No other relationships are known. A Venn diagram of these relationships is below:

Assume both statements to be true. The question is whether or not Jack falls in set . If Jack is neither a Fi nor a Fo, then Jack falls in neither set  nor set . As evidenced by the two X's in the diagram, it is possible for Jack to fall in set  or to not fall in set . Therefore, the question of whether Jack is a Fum is unresolved.

Let  represent the sets of people who like apples, bananas, etc. , respectively. Let  represent Smith.

Assume Statement 1 alone. 

If , then . Contrapositively, if  , then . Therefore,  cannot be in both  and ;  is in one, the other, or neither. By similar reasoning,  is in at most one of  and  and at most one of  and . 

Suppose  and . Then  is in three of , and either  falls in both  and  or both  and ; since both are impossible, it follows that  or , but not both. Therefore, Smith likes exactly one of bananas and figs, but not both. Since, from Statement 1, Smith doesn't like bananas, he must like figs.

Assume Statement 2. By similar reasoning, Smith likes cranberries or dates, but not both. Since he does not like cranberries, he likes dates. However, without further information, it cannot be determined what else he likes or dislikes.

Assume Statement 1 alone. Let  represent the sets of Beeps, Deeps, Meeps, and Veeps. Then  and, since all Veeps are not Deeps, . This is represented by the Venn diagram below:

Note that the set  (Meeps) need not fall completely inside the set  (Veeps). Therefore, Harpy being a Meep does not necessarily make him a Veep or not a Veep.

Assume Statement 2 alone. Every Beep is a Deep, but no Veep is a Deep. If Harpy is a Beep, then he is a Deep. If Harpy were a Veep, then he cannot be a Deep, so, by contradiction, Harpy cannot be a Veep.

Assume Statement 1 alone.  is the intersection of the complements of all three of , , and  - the sets of people who do not like Neil Young, people who do not like Prince, and people who do not like Marvin Gaye. Any person who is in this intersection does not like any of these three. , so Phillip does not like Neil Young, Prince, or Marvin Gaye.

Assume Statement 2 alone.  is the union of all three , , and . Any person who is in this union likes any one, two, or three of these musicians. However, , which is the complement of this union - the set of people who like none of Neil Young, Prince, or Marvin Gaye. Phil is in this set.