By Patrick Murphy

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5. If London is in France then Paris is in England. 6. If New York is in the USA then Paris is in Australia. 7. If sets A, Β are disjoint (T) then AnB = AuB. 8. If X # 0 and X < 0 then χ > 0. 9. For a quadrilateral to be a square it is sufficient that all of the angles be right-angles. 10. A necessary condition for a quadrilateral to be a square is that all its sides be of equal length. Bicoiiditioiuil Sentences As may be expected by the name, a biconditional χ y is an implication which is read two ways—^that is, not only does χ imply y but y implies x.

For example, when we say Tor A and b, any two positive numbers < IR if and only IF A < b\ we mean (i) Ifa^ < b^ thena < b, as well as (ii) Iffl < i^thenö^ < b\ Therefore, to prove a biconditional will require not only the proof of a theorem but also a proof of the converse theorem. Now, with any theorem χ follows: (i) (ii) (iii) (iv) Theorem: Converse: Inverse: Contrapositive: χ y χ y y we associate three main ideas which we list as y χ y χ We shall now explore some of the relations between these ideas.

7 and simplify the circuit if possible. R -4Fig. 7 Solutioo. Here we have two series circuits PQS and PRS wired in parallel. y. z. z By inspection we can see that if χ = 0 then / = 0 and this suggests that we need only use one X switch in the simpler circuit of Fig. 8 to obtain the same flow function. In this simpler circuit we have circuit FT in series with circuit TS f - x,(y •\- ζ) is the flow function Ρ Fig. z In other words, we may distribute . over + to obtain the result on the right-hand side (see page 19 with .