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Proofs 24 I / The Foundations: Logic and TABLf, 6 Logical Equival � nces. r q ) v r = p v \ q \ t r ) \ p ( p ^ q ) ^ / = p ^ l q ^ t ) ^ /) = (p v tt) ^ \? v r) p v (.1 ^ q)v \1, p ^ (r1't /) = Q, ^t) * In these rhe c-( equivalences' T denotes important equivalences 6 contains some Table the compound proposinon thal t5 pouild proposition thai is always true and F denotes proPosinons itnolr useful equivalences for compound ways false. We also display some 8, respectivev Th � r � ' ? and biconditional statemenis in Tables conditional statements and at the end ofthe s � cnoo_ to veriry the equialences in Tables 6-8 in the exercises is asked P v q v r is u � ll defit law lor disjunction sho{'s that the expression The associative we fust take the disjunction of p with g and t not matter whether in the sense tha! it does / ard then Bte the disjunction of 4 and the disjunction ofp v q with /, or ifwe frrst take P ^ 4 ,\ r'is well defned By er(renditrg disjunition ofp with q v r. Similarly, the expression ptv p2\r " vr-arrdpi sile r /\p' are well deined it f;llows that rcaionhg, P2A lo that De Motgant laws extend p t, p2, . ., p, Ne prcpositions. Furthermore, note -(p1\t p2't ..v pn)= (-Pr A-p2A ^-?,) and v-?,). -(pr r, p2 /\.. ^p.)=(-prv-p,v identities $'ill be given h Section 4 l ) (Methods for proving these " lde.tilies c a specitl G of ij.tu llgetB will noticc lhat rhese Reades fmillar wilh the concept of a Eoolean rYith 8@16 id'ndc 6 in T.ble I in Section 2 2 and Boolem algebn. ConPde dlen wilh set identities hold fot any

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