1 Consider the language L := {c, R2} consisting of a constant and a binary relation symbol. a Find two non-isomorphic L -structures.
b Prove that the number of symbols in any L -formula is divisible by 3.
2 For the language L := {c,f1,g2,R3} consisting of one constant, one unary function symbol, one binary function symbol, and one ternary relation symbol, define an L -structure on the set of L -terms in a way such that the interpretation of a term t is not equal to t.
3 Let L be a first-order language and let A be an L -structure. If s is a variable assignment function into A , prove that A |= (\u2203x)(\u03b1)[s] if, and only if, there exists some a \u2208 A such that A |= \u03b1[s[x|a]].
4 You might think that (\u03c6xy)yx is equal to \u03c6, but a moment\u2019s thought will give you an example to show that this doesn\u2019t always work. (What if y is free in \u03c6?) Find an example that shows that even if y is not free in \u03c6, we can still have that (\u03c6xy)yx is different from \u03c6. Under what conditions do we know that (\u03c6xy )yx is equal to \u03c6?
5 Let \u03c6 and \u03c8 be L -formulas.
(a) Prove that if |= (\u03c6 \u2192 \u03c8) then \u03c6 |= \u03c8.
(b) If\u03c6:=x<y,and\u03c8:=z<w,provethat\u03c6|=\u03c8but\u0338|=(\u03c6\u2192\u03c8).
This exercise shows that the two possible ways to define logical equivalence are not equivalent <\/p>\n","protected":false},"excerpt":{"rendered":"
1 Consider the language L := {c, R2} consisting of a constant and a binary relation symbol. a Find two non-isomorphic L -structures. b Prove that the number of symbols in any L -formula is divisible by 3. 2 For the language L := {c,f1,g2,R3} consisting of one constant, one unary function symbol, one binary […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[33],"class_list":["post-52615","post","type-post","status-publish","format-standard","hentry","category-research-paper-writing","tag-mathematics"],"_links":{"self":[{"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/posts\/52615","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/comments?post=52615"}],"version-history":[{"count":0,"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/posts\/52615\/revisions"}],"wp:attachment":[{"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/media?parent=52615"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/categories?post=52615"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/papersspot.com\/blog\/wp-json\/wp\/v2\/tags?post=52615"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}