### Conditional Connectives

• For two propositions p and q, we denote the conditional statement "If p then q" by "p → q". p → q is also read as "p implies q" or "p is sufficient for q" or "p only if q". p is called the hypotheses and q is called the conclusion. It is also called conditional proposition . Also known as implication.

The compound proposition "If the bus comes then I will go to the market." is made up of "p: The bus comes" and "q: I will go to the market". This compound proposition can be written as "If p then q" or in mathematical term "p → q".

The proposition is also an implication of q by p.
Truth table for the Implication

pqp → q
TTT
TFF
FTT
FFT

• The statement "p → q ∧ q → p" is a biconditional statement and is denoted as "p ↔ q". "p ↔ q" is read as "p if and only if q". It is also read as "p implies and is implied by q" or "p is necessary and sufficient for q".

The proposition "p: The bus comes" and "q: I will go to the market" if written as q ↔ p implies "I will go to the market if and only if the bus comes."
Truth table for the two way Implication

pqp → qq → pp ↔ q
TTTTT
TFFTF
FTTFF
FFTTT