Logical connectives (real analysis)

Declarative sentence

According to Wikipedia, a declarative sentence is a type of a sentence that commonly makes a statement. For example I am going home. A statement is a sentence that can be classified as true or false. Every statement, thus, has a truth value (true or false and never both).

Logical connectives 

Logical connectives connect multiple statements. For example, 

There is little cloud cover. Today is a warm day.

These two sentences can be combined in a variety of ways. Let us look at some of them: 

1. There is little cloud cover and today is a warm day. 
2. There is little cloud cover or today is a warm day.
3. If there is little cloud cover, then today is a warm day. 
4. Today is a warm day if and only if there is little cloud cover. 

Sentential connectives: Words such as and, or, not, if ... then, if and only if are called sentential connectives.  Let us look at each in brief. 

And: An and statement is true only when both component sentences in the and sentence are true.  In other words, it is false when any of the components is false. The symbol used for and is ^. It can be read as hat. We shall use the word cap though. 

Or: An or statement is true when at least one of the component sentences is true. In other words, it is false only when all the components are false. The symbol used for or is something like a v.

Not: A negation flips the truth value of a statement. The symbol for not is ¬. If you are reading ¬p, you would read it as "not p". 

If ... then: An implication p→q is logically equivalent to ¬(p^¬q). p→q can be read as if p then q.